Notes

Bioprocess Engineering — Revision sheet

Every formula and key point across this section, in one scroll — for the daily pass, not the first read.

1.0

Fluid Mechanics

Key points

01
Ideal fluids are inviscid and incompressible (a hypothetical simplification); real fluids are viscous — compressibility is a separate, independent property.
02
Newton's law of viscosity ( $\tau = \mu\dot\gamma$ ) only holds for Newtonian fluids, where $\mu$ is constant; non-Newtonian fluids need an apparent viscosity ( $\mu_a = K\dot\gamma^{\,n-1}$ ) that varies with shear rate instead.
03
Flow behaviour index $n$ classifies non-Newtonian fluids: $n < 1$ pseudoplastic (shear-thinning), $n > 1$ dilatant (shear-thickening).
04
Bingham plastics need a yield stress $\tau_y$ before they flow at all; Casson and Herschel–Bulkley fluids combine a yield stress with shear-thinning/power-law behaviour above it.
05
Non-Newtonian fluids also split by time-dependence: Bingham, pseudoplastic, dilatant, Casson, and Herschel–Bulkley are all time-independent (viscosity depends only on instantaneous shear rate); thixotropic fluids thin over time at constant shear rate, rheopectic fluids thicken over time at constant shear rate.
06
Reynolds number thresholds differ by system: stirred tank is laminar below $Re = 10$ and turbulent above $Re = 10{,}000$ ; straight pipe is laminar below $Re = 2{,}100$ and turbulent above $Re = 4{,}000$ — don't cross-apply the two.
07
Turbulent flow (with eddies) is needed for mixing; laminar flow is not, and neither is a vortex — a vortex is easy to mistake for turbulence but actually promotes settling, not mixing.
08
Laminar pipe flow gives a parabolic velocity profile (peak at centreline, zero at wall); turbulent pipe flow flattens it into a near-plug flow shape, with the steep drop confined to a thin near-wall layer.
09
Human blood's Casson-type yield stress comes from rouleaux formation — red blood cells stacking at low shear rates — not from a generic "blood is thick" explanation.

Shear stress

\tau = \frac{F}{A}

$\tau$ — shear stress (tangential force per unit area, resisting relative sliding between adjacent fluid layers), in pascals ( $\text{Pa}$ ) or newtons per square metre ( $\text{N/m}^2$ )
$F$ — tangential force, in newtons ( $\text{N}$ )
$A$ — area, in square metres ( $\text{m}^2$ )

Shear rate

\dot{\gamma} = \frac{v}{h}

$\dot{\gamma}$ — shear rate (velocity gradient), in reciprocal seconds ( $\text{s}^{-1}$ )
$v$ — velocity of the topmost fluid layer, in metres per second ( $\text{m/s}$ )
$h$ — height of the fluid layer, in metres ( $\text{m}$ )

Kinematic viscosity

\nu = \frac{\mu}{\rho}

$\nu$ — kinematic viscosity (momentum diffusivity — viscous effect on the fluid, independent of its mass or force), in square metres per second ( $\text{m}^2/\text{s}$ )
$\mu$ — dynamic viscosity, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )

Power law equation (Ostwald–de Waele model)

\tau = K \dot{\gamma}^n

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$K$ — consistency index (generalized viscosity), in $\text{Pa} \cdot \text{s}^n$ (or $\text{N} \cdot \text{s}^n / \text{m}^2$ ): $K \uparrow \rightarrow$ viscosity $\uparrow$ ; $K \downarrow \rightarrow$ viscosity $\downarrow$
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )
$n$ — flow behaviour index (exponent that classifies shear-thinning vs. shear-thickening behaviour), dimensionless

Apparent viscosity of a power-law fluid

\mu_a = K \dot{\gamma}^{\,n-1}

$\mu_a$ — apparent viscosity (the effective, shear-rate-dependent viscosity of a non-Newtonian fluid at an instant — stands in for constant $\mu$ in equations built for Newtonian fluids), in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$K$ — consistency index, in $\text{Pa} \cdot \text{s}^n$
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )
$n$ — flow behaviour index (exponent that classifies shear-thinning vs. shear-thickening behaviour), dimensionless

Newton's law of viscosity

\tau = \mu \dot{\gamma} \quad \text{or} \quad \frac{\tau}{\dot{\gamma}} = \mu

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\mu$ — dynamic viscosity (constant for a Newtonian fluid — independent of the shear stress or shear rate applied), in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )

Bingham plastics

\tau = \tau_y + \mu_p \dot{\gamma}

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\tau_y$ — yield stress (threshold stress that must be exceeded before the material begins to flow at all), in pascals ( $\text{Pa}$ )
$\mu_p$ — plastic viscosity (also called apparent viscosity), in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )

Casson plastics

\sqrt{\tau} = \sqrt{\tau_0} + \sqrt{K_c \dot{\gamma}}

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\tau_0$ — Casson yield stress (threshold stress below which the fluid behaves like a Bingham plastic), in pascals ( $\text{Pa}$ )
$K_c$ — Casson viscosity coefficient, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )

Herschel–Bulkley model

\tau = \tau_y + K \dot{\gamma}^n

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\tau_y$ — yield stress (threshold stress that must be exceeded before the material begins to flow at all), in pascals ( $\text{Pa}$ )
$K$ — consistency index, in $\text{Pa} \cdot \text{s}^n$
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )
$n$ — flow behaviour index (exponent that classifies shear-thinning vs. shear-thickening behaviour), dimensionless

Reynolds number for stirred tank

Very important

Re = \frac{d_i^2 N_i \rho}{\mu}

$Re$ — Reynolds number (ratio of inertial to viscous forces), dimensionless
$d_i$ — diameter of the impeller, in metres ( $\text{m}$ )
$N_i$ — frequency of the impeller, in reciprocal seconds ( $\text{s}^{-1}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )
$\mu$ — viscosity, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$Re < 10$ — laminar
$10 \le Re \le 10{,}000$ — transitional
$Re > 10{,}000$ — turbulent

Reynolds number for straight pipe

Re = \frac{d v \rho}{\mu}

$Re$ — Reynolds number (ratio of inertial to viscous forces), dimensionless
$d$ — diameter of the pipe, in metres ( $\text{m}$ )
$v$ — average velocity of the fluid, in metres per second ( $\text{m/s}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )
$\mu$ — viscosity, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$Re < 2{,}100$ — laminar
$Re > 4{,}000$ — turbulent

Froude number for stirred tank

Fr = \frac{N_i^2 d_i}{g}

$Fr$ — Froude number (ratio of inertial to gravitational/buoyancy forces), dimensionless
$N_i$ — frequency of the impeller, in reciprocal seconds ( $\text{s}^{-1}$ )
$d_i$ — diameter of the impeller, in metres ( $\text{m}$ )
$g$ — acceleration due to gravity, $9.8 \ \text{m/s}^2$

Full note →

2.0

Mixing

Key points

01
Turbulence is a spectrum of eddy sizes: the impeller feeds kinetic energy into the largest eddies, it cascades down to progressively smaller ones, and viscous dissipation into heat happens almost entirely at the smallest (Kolmogorov microscale) eddies.
02
Shear damage is worst near the impeller blades, where the smallest, most energetic eddies form — this is what fragments filamentous fungal hyphae and lyses wall-less animal cells.
03
Mixing time should be minimized — a slow mix lets local pH or concentration spikes near the addition point kill nearby cells.
04
Baffles break circulatory flow and vortex formation, converting it into turbulence and eddies; a standard reactor uses 4.
05
Power number ( $N_p$ ) is constant only in turbulent flow — in laminar flow it depends on Reynolds number, so always check the flow regime first.
06
Flow number ( $F_l$ ) scales with $D_i^3$ while power number scales with $D_i^5$ — the two formulas look alike but respond very differently to impeller size.
07
Radial (Rushton) impellers give high shear and high $O_2$ transfer; axial impellers (propeller, hydrofoil, pitched blade) give low-to-moderate shear and high pumping — never use Rushton for shear-sensitive cultures.
08
Viscosity ladder for impeller choice: pitched blade → anchor → helical ribbon, as fluids get thicker and more non-Newtonian; anchor is the only one built to scrape the tank wall.

Power number (Newton number, $N_p$ )

N_p = \frac{P}{\rho N_i^3 D_i^5}

$N_p$ — power number (Newton number — dimensionless power drawn to agitate the fluid at a given speed and impeller diameter, analogous to a drag coefficient), dimensionless
$P$ — power, in watts ( $\text{W}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

Power in laminar flow

P = K_L \mu N_i^2 D_i^3

$P$ — power, in watts ( $\text{W}$ )
$K_L$ — laminar power constant, a dimensionless constant depending on the geometry of the impeller and tank (blade pitch, blade shape, number of blades, vessel geometry)
$\mu$ — dynamic viscosity of the fluid, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

Impeller tip speed

v_{tip} = \pi N_i D_i

$v_{tip}$ — impeller tip speed (maximum linear velocity at the blade edge — a proxy for the shear intensity cells experience near the impeller), in metres per second ( $\text{m/s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

Flow number ( $F_l$ / $F_Q$ )

F_l = \frac{Q}{N_i D_i^3}

$F_l$ — flow number (also written $F_Q$ — dimensionless measure of pumping capacity: how much fluid volume the impeller circulates per revolution), dimensionless
$Q$ — volumetric flow rate of fluid leaving the impeller blade, in cubic metres per second ( $\text{m}^3/\text{s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

If power is given

t_m = 5.9\, D_T^{2/3} \left(\frac{\rho V_L}{P}\right)^{1/3} \left(\frac{D_T}{D_i}\right)^{1/3}

$t_m$ — mixing time (time required for an added pulse of tracer/chemical to become homogeneously distributed through the vessel), in seconds ( $\text{s}$ )
$D_T$ — tank diameter, in metres ( $\text{m}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )
$V_L$ — volume of liquid, in cubic metres ( $\text{m}^3$ )
$P$ — power, in watts ( $\text{W}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

If power number is given

t_m = \frac{5.4}{N_i} \left(\frac{1}{N_p}\right)^{1/3} \left(\frac{D_T}{D_i}\right)^2

$t_m$ — mixing time (time required for an added pulse of tracer/chemical to become homogeneously distributed through the vessel), in seconds ( $\text{s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$N_p$ — power number (dimensionless power drawn to agitate the fluid), dimensionless
$D_T$ — tank diameter, in metres ( $\text{m}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

Full note →

3.0

Mass Transfer

Key points

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.
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).
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.
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.
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.
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.
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).
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.

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)

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

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 ( $\varepsilon_g$ )

\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 ( $\varepsilon_g$ )

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

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

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

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 $C^*_{AL}$

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

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

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)

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

Full note →

4.0

Heat Transfer

Key points

01
Conduction moves heat via vibrational energy or free electrons, with no movement of matter; the sign on Fourier's law only tracks direction — $\hat{Q}$ itself always comes out positive.
02
Heat crossing a jacketed wall moves in three steps: convection (bulk fluid to wall), conduction (through the wall), convection (wall to bulk fluid) — driven by two separate $\Delta T$ s, one per convective step.
03
Nusselt number ( $Nu = hD/k_{fb}$ ) is the convective-to-conductive heat transfer ratio; Prandtl number ( $Pr = C_p\mu_b/k_{fb}$ ) is the momentum-diffusivity-to-thermal-diffusivity ratio — the calculation chain runs $Pr \to Nu \to h \to \hat{Q}$ , backwards from what you actually want.
04
Overall heat transfer coefficient $U$ avoids measuring $\Delta T$ at each wall interface: $1/U = 1/h_1 + x/k + 1/h_2$ combines both film coefficients with the wall's own conduction resistance.
05
$\Delta T_{LMTD} = (\Delta T_2 - \Delta T_1)/\ln(\Delta T_2/\Delta T_1)$ . For fixed terminal temperatures, $\Delta T_1 + \Delta T_2$ is identical for parallel and counter-current flow, but counter-current keeps $\Delta T_1$ and $\Delta T_2$ closer together — so its LMTD is always $\geq$ parallel flow's, never smaller.
06
The two arrangements only tie when one fluid doesn't change temperature at all (a condensing vapor or boiling liquid) — otherwise counter-current always wins.
07
Metabolic heat tracks oxygen consumption — Cooney's correlation: $\approx 460\,\text{kJ/mol}\ O_2$ ( $\approx 14\,\text{kJ/g}\ O_2$ ).
08
Scale-up hurts cooling because $V \propto D^3$ but cooling area $A \propto D^2$ , so $A/V \propto 1/D$ ; external jackets stay the default (and are preferred for shear-sensitive cultures), while internal coils add area at large scale at the cost of extra shear and working volume.

Rate of heat transfer by conduction

\hat{Q} = -kA\frac{dT}{dy} = -kA\frac{T_2 - T_1}{y}

$\hat{Q}$ — rate of heat transfer (thermal energy crossing area $A$ per unit time), in watts ( $\text{W}$ )
$k$ — thermal conductivity of the wall, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )
$A$ — area perpendicular to heat transfer, in square metres ( $\text{m}^2$ )
$dT/dy$ — temperature gradient, in kelvin per metre ( $\text{K/m}$ )
$T_2, T_1$ — temperatures either side of the wall, in kelvin ( $\text{K}$ )
$y$ — distance component of thickness, in metres ( $\text{m}$ )

Rate of heat transfer by convection

\hat{Q} = hA \Delta T

$\hat{Q}$ — rate of heat transfer (thermal energy crossing area $A$ per unit time), in watts ( $\text{W}$ )
$h$ — individual (film) heat transfer coefficient — how readily heat crosses between the bulk fluid and the wall by convection, in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$A$ — area, in square metres ( $\text{m}^2$ )
$\Delta T$ — temperature gradient, in kelvin ( $\text{K}$ )

Nusselt number

Nu = \frac{hD}{k_{fb}}

$Nu$ — Nusselt number (ratio of convective to conductive heat transfer), dimensionless
$h$ — individual heat transfer coefficient, in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$D$ — diameter of the pipe or tank, or characteristic length of the pipe, in metres ( $\text{m}$ )
$k_{fb}$ — thermal conductivity of the bulk fluid, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )

Prandtl number

Pr = \frac{C_p \mu_b}{k_{fb}}

$Pr$ — Prandtl number (ratio of molecular momentum to thermal diffusivity), dimensionless
$C_p$ — heat capacity, in joules per kilogram-kelvin ( $\text{J}/(\text{kg} \cdot \text{K})$ )
$\mu_b$ — viscosity of the bulk fluid, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$k_{fb}$ — thermal conductivity of the bulk fluid, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )

Conductive resistance of a wall

R_{cond} = \frac{x}{k}

$R_{cond}$ — conductive thermal resistance per unit area, in square metre-kelvin per watt ( $\text{m}^2 \cdot \text{K}/\text{W}$ )
$x$ — thickness of the layer, in metres ( $\text{m}$ )
$k$ — thermal conductivity of the layer's material, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )

Conductive resistance of a wall

R_{total} = \sum_i \frac{x_i}{k_i}

$R_{total}$ — total conductive resistance of the composite wall per unit area, in square metre-kelvin per watt ( $\text{m}^2 \cdot \text{K}/\text{W}$ )
$x_i, k_i$ — thickness and thermal conductivity of layer $i$ , in metres ( $\text{m}$ ) and watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ ) respectively

Rate of heat transfer using U

\hat{Q} = UA \Delta T_{LMTD}

$\hat{Q}$ — rate of heat transfer (thermal energy crossing area $A$ per unit time), in watts ( $\text{W}$ )
$U$ — overall heat transfer coefficient (lumps both film coefficients and the wall's own conduction resistance into one number, avoiding the need to measure $\Delta T$ at each internal interface), in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$A$ — area, in square metres ( $\text{m}^2$ )
$\Delta T_{LMTD}$ — logarithmic mean temperature difference, in kelvin ( $\text{K}$ )

Overall heat transfer coefficient from film coefficients

\frac{1}{U} = \frac{1}{h_1} + \frac{x}{k} + \frac{1}{h_2} + R_f

$U$ — overall heat transfer coefficient (lumps both film coefficients and the wall's own conduction resistance into one number), in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$h_1, h_2$ — individual film heat transfer coefficients on either side of the wall, in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$x$ — thickness of the wall, in metres ( $\text{m}$ )
$k$ — thermal conductivity of the wall, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )
$R_f$ — fouling resistance at a film surface (e.g. microbial biofilm or scale buildup), in square metre-kelvin per watt ( $\text{m}^2 \cdot \text{K}/\text{W}$ )

Logarithmic mean temperature difference (LMTD)

\Delta T_{LMTD} = \frac{\Delta T_2 - \Delta T_1}{\ln(\Delta T_2 / \Delta T_1)}

$\Delta T_{LMTD}$ — logarithmic mean temperature difference (single equivalent driving-force temperature for the whole exchanger, replacing the $\Delta T$ that actually varies along its length), in kelvin ( $\text{K}$ )
$\Delta T_2, \Delta T_1$ — temperature differences at the two ends of the heat exchanger, in kelvin ( $\text{K}$ )

Metabolic heat from oxygen consumption (Cooney's correlation)

Q_{metabolic} = \Delta H_{O_2} \times n_{O_2}

$Q_{metabolic}$ — metabolic heat released by aerobic respiration, in kilojoules ( $\text{kJ}$ )
$\Delta H_{O_2}$ — heat released per mole of oxygen consumed (**Cooney's correlation**), approximately $460\,\text{kJ/mol}$ ( $\approx 14\,\text{kJ/g}$ $O_2$ , since $460/32 \approx 14.4$ )
$n_{O_2}$ — moles of oxygen consumed by the culture ( $\text{mol}$ )

Scale-up heat transfer challenge

\frac{A}{V} \propto \frac{D^2}{D^3} = \frac{1}{D}

$A$ — cooling surface area available (jacket or coil), in square metres ( $\text{m}^2$ )
$V$ — reactor working volume, in cubic metres ( $\text{m}^3$ )
$D$ — characteristic vessel diameter, in metres ( $\text{m}$ )

Full note →

5.0

Recycle, Purge, and Bypass

Key points

01
Recycle returns part of the product stream to the feed to increase residence time and overall conversion — but pushed too far, it can reduce productivity.
02
Recycle raises overall conversion of fresh feed, not necessarily single-pass conversion — per-pass performance typically drops toward CSTR-like behavior as the recycle ratio grows.
03
Cell recycle decouples hydraulic retention time ( $1/D$ ) from biomass retention time — returning cells that would otherwise wash out lets a bioreactor run at a dilution rate above $\mu_{max}$ without washing out.
04
Bypass diverts part of the feed around the process and rejoins it with the product, used when the product is too concentrated or when temperature control is needed for an exothermic process.
05
Purge bleeds off part of the recycle stream so inerts or impurities entering with the feed, or forming inside the reactor, don't build up.

Dilution rate, HRT, and washout

D = \frac{F}{V} \qquad\qquad \text{HRT} = \frac{V}{F} = \frac{1}{D}

$D$ — dilution rate (how fast the reactor volume is being flushed and replaced with fresh feed), $\text{h}^{-1}$
$F$ — feed flow rate, $\text{L/h}$
$V$ — reactor (working) volume, $\text{L}$
$\text{HRT}$ — hydraulic retention time (average time a parcel of liquid spends in the reactor before it flows out — pure plumbing, no biology involved), $\text{h}$

Full note →

6.0

Cell Stoichiometry

Key points

01
The general growth equation has five unknown coefficients but only four elemental balances (C, H, O, N) — a fifth relation (measured $RQ$ , measured yield, or an electron balance) is always needed to solve it.
02
If a nitrogen-containing carbon source supplies more nitrogen than the biomass formed needs, the nitrogen-source coefficient $b$ comes out negative — meaning $NH_3$ is a product, not a reactant.
03
Respiratory quotient $RQ = \dfrac{\text{mol } CO_2}{\text{mol } O_2}$ : $\approx 1$ for carbohydrates, $< 1$ for reduced substrates (fats, hydrocarbons), $> 1$ for already-oxidized substrates (organic acids, amino acids).
04
Available electrons = the sum of (valency × atom count) for every element in a formula; $C=+4$ , $H=+1$ , $O=-2$ , $P=+5$ , $S=+6$ are fixed, but $N$ 's valency depends on the compound it's found in.
05
Degree of reduction $\gamma$ = available electrons ÷ number of carbon atoms — it measures how reduced a substance is per carbon, and is what lets an electron balance replace an elemental balance when solving the general growth equation. $CO_2$ is the zero-point ( $\gamma=0$ ); $CH_4$ is the highest single-carbon value ( $\gamma=8$ ).
06
One mole of $O_2$ = $4$ electron equivalents — the conversion used to turn "electrons left over" into moles of $O_2$ in any electron balance, and it follows directly from $O_2 + 4H^+ + 4e^- \rightarrow 2H_2O$ .
07
$\gamma_X$ (biomass) doesn't depend on the nitrogen source — it's fixed by biomass's elemental composition, using amine-nitrogen ( $-3$ ) valency, because that's the level nitrogen sits at once it's inside the cell either way. Feeding $NO_3^-$ instead of $NH_3$ changes the electron balance instead (an extra $8$ electrons per N atom to reduce nitrate down to the amine level), which shows up as lower $O_2$ demand, not a lower $\gamma_X$ .
08
A biomass elemental formula is written per C-mole (one carbon atom per formula unit), so mass-percent composition is just each element's (subscript × atomic weight) over the formula's total weight.
09
Product formation adds a sixth term to the general equation ( $f\,C_jH_kO_lN_m$ ) and a matching term to the electron balance ( $\gamma_P j f$ ) — the product's electron total needs the $\times j$ because, unlike biomass, it isn't written on a C-mole basis.
10
Maximum theoretical yield ( $Y_{X/S}^{max}$ , $Y_{P/S}^{max}$ ) is the electron balance with $O_2$ zeroed out — all substrate electrons land in biomass or product instead. Real aerobic cultures always fall below it, since some electrons are always spent on $O_2$ .
11
Theoretical oxygen demand (ThOD) is the same electron balance run the other way: $c = f = 0$ , every electron goes to $O_2$ — the $O_2$ needed to fully combust the substance.

General equation of aerobic growth

C_wH_xO_yN_z + aO_2 + bH_gO_hN_i \rightarrow cCH_\alpha O_\beta N_\delta + dCO_2 + eH_2O

$w, x, y, z$ — subscripts of the carbon source's molecular formula
$a$ — stoichiometric coefficient of oxygen (mol)
$b$ — stoichiometric coefficient of the nitrogen source (mol)
$g, h, i$ — subscripts of the nitrogen source's molecular formula
$c$ — stoichiometric coefficient of biomass (mol)
$\alpha, \beta, \delta$ — subscripts of the biomass formula, on a C-mole basis (one carbon atom per formula unit)
$d$ — stoichiometric coefficient of carbon dioxide (mol)
$e$ — stoichiometric coefficient of water (mol)

Respiratory quotient (RQ)

RQ = \frac{\text{no. of moles of } CO_2 \text{ produced}}{\text{no. of moles of } O_2 \text{ consumed}}

$RQ$ — respiratory quotient (ratio of CO $_2$ produced to O $_2$ consumed — indicates how reduced or oxidized the substrate is: a **reduced** substrate is rich in hydrogen relative to oxygen (fats, hydrocarbons — still holds a lot of energy/available electrons, so needs a lot of $O_2$ to fully combust), while an **oxidized** substrate already has more oxygen bonded in and fewer available electrons left to give up (organic acids, and $CO_2$ at the fully-oxidized extreme) — see degree of reduction, below, for the precise measure of this), dimensionless

Degree of reduction ( $\gamma$ )

\gamma = \frac{\text{no. of available electrons}}{\text{no. of carbon atoms}}

$\gamma$ — degree of reduction (available electrons per carbon atom — measures how reduced/energy-rich a substance is; higher $\gamma$ needs more $O_2$ to fully oxidize), dimensionless

Electron yields ( $\eta$ )

\eta_X = \frac{\gamma_X c}{\gamma_S w} \qquad\qquad \eta_{O_2} = \frac{4a}{\gamma_S w}

$\eta_X$ — fraction of substrate electrons transferred to biomass, dimensionless
$\eta_{O_2}$ — fraction of substrate electrons transferred to $O_2$ , dimensionless
$\gamma_S w$ — total available electrons entering with the substrate (denominator for every $\eta$ )

General equation with product formation

C_wH_xO_yN_z + aO_2 + bH_gO_hN_i \rightarrow cCH_\alpha O_\beta N_\delta + fC_jH_kO_lN_m + dCO_2 + eH_2O

$f$ — stoichiometric coefficient of the product (mol)
$j, k, l, m$ — subscripts of the product's molecular formula — written as the real molecule (e.g. $C_2H_6O$ for ethanol), not on a C-mole basis the way biomass is

Yield coefficients

Y_{X/S} = \frac{\Delta X}{\Delta S} \qquad\qquad Y_{P/S} = \frac{\Delta P}{\Delta S}

$Y_{X/S}$ — biomass yield on substrate (fraction of substrate mass converted into biomass), mass basis, in grams of biomass per gram of substrate consumed ( $\text{g/g}$ )
$Y_{P/S}$ — product yield on substrate (fraction of substrate mass converted into product), mass basis, in grams of product per gram of substrate consumed ( $\text{g/g}$ )
$\Delta X$ , $\Delta P$ , $\Delta S$ — biomass formed, product formed, and substrate consumed, in grams ( $\text{g}$ )

Theoretical oxygen demand (ThOD)

\text{ThOD} = \frac{\text{available electrons per mole}}{4}

$\text{ThOD}$ — theoretical oxygen demand ( $O_2$ needed to fully combust the substance — the ceiling on $O_2$ consumption, reached when no electrons go to biomass or product), in moles of $O_2$ per mole of substance ( $\text{mol } O_2/\text{mol}$ ); multiply by $32\ \text{g/mol}$ and divide by the molar mass for a mass basis ( $\text{g } O_2/\text{g}$ )

Elemental composition of biomass

\%\,\text{element}_i = \frac{n_i M_i}{\sum_j n_j M_j} \times 100

$\%\,\text{element}_i$ — mass contribution of element $i$ to dry biomass (what fraction of the cell's dry weight that element accounts for), percent ( $\%$ )
$n_i$ — subscript (atom count) of element $i$ in the biomass formula, dimensionless
$M_i$ — atomic weight of element $i$ , in grams per mole ( $\text{g/mol}$ )
$\sum_j n_j M_j$ — sum of (subscript × atomic weight) over every element in the formula, in grams per mole ( $\text{g/mol}$ ) — the total C-mole formula weight

Full note →

7.0

Thermodynamics

Force–displacement form

W = Fd

$W$ — work (energy transferred by a force acting through a displacement), joule (J)
$F$ — force, newton (N)
$d$ — displacement, metre (m)

Pressure–volume form

W = P\Delta V

$W$ — work (energy transferred by expansion or compression against an external pressure), joule (J)
$P$ — pressure, pascal (Pa)
$\Delta V$ — change in volume, cubic metre (m³)

Internal energy

\Delta U = U_2 - U_1

$\Delta U$ — change in internal energy (net energy gained or lost by the system as heat and work cross its boundary), joule (J)
$U_2$ — final internal energy of the system, joule (J)
$U_1$ — initial internal energy of the system, joule (J)

Internal energy

\Delta U = \pm q \pm w

$q$ — heat transferred across the system boundary, joule (J)
$w$ — work transferred across the system boundary, joule (J)

Enthalpy

\Delta H = \Delta U + P\Delta V

$\Delta H$ — change in enthalpy (the heat exchanged at constant pressure — internal energy minus the boundary-pushing $P\Delta V$ work not felt as heat), joule (J)
$\Delta U$ — change in internal energy, joule (J)
$P$ — pressure, pascal (Pa)
$\Delta V$ — change in volume, cubic metre (m³)

Second law of thermodynamics

\Delta S = S_2 - S_1

$\Delta S$ — change in entropy (change in the system's disorder — positive for any spontaneous, irreversible process), joule per kelvin (J/K)
$S_2$ — final entropy of the system, joule per kelvin (J/K)
$S_1$ — initial entropy of the system, joule per kelvin (J/K)

Entropy of the surroundings

\Delta S_{surr} = -\dfrac{\Delta H_{sys}}{T_{surr}}

$\Delta S_{surr}$ — change in entropy of the surroundings, joule per kelvin (J/K)
$\Delta H_{sys}$ — change in enthalpy of the system, joule (J)
$T_{surr}$ — temperature of the surroundings, kelvin (K)

Gibbs free energy

\Delta G = \Delta H - T\Delta S

$\Delta G$ — change in Gibbs free energy, joule (J)
$\Delta H$ — change in enthalpy, joule (J)
$T$ — absolute temperature, kelvin (K)
$\Delta S$ — change in entropy, joule per kelvin (J/K)

Full note →

8.0

Ligand Binding and the Scatchard Plot

Key points

01
$K_a = 1/K_d$ : a high $K_d$ means low binding affinity; a high $K_a$ means high binding affinity.
02
The Scatchard plot linearizes saturation binding data: plot $r/[L]$ (or $\Theta/[L]$ ) against $r$ (or $\Theta$ ) to read off affinity and binding-site count without curve-fitting.
03
Slope $= -K_a$ , y-intercept $= nK_a$ , x-intercept $= n$ .
04
$\Theta = r/n$ is fractional saturation, always between 0 and 1; $r$ itself is not bounded by 1 for a multi-site protein.
05
For a protein with a single binding site per molecule ( $n = 1$ ), total binding-site concentration equals total protein concentration.

Ligand binding equilibrium

[P] + [L] \underset{k_d}{\overset{k_a}{\rightleftharpoons}} [PL]

$[P]$ : free protein concentration, $\text{M}$
$[L]$ : free ligand concentration, $\text{M}$
$[PL]$ : protein-ligand complex concentration, $\text{M}$
$k_a$ : rate constant for complex formation (association), $\text{M}^{-1}\text{s}^{-1}$
$k_d$ : rate constant for complex breakdown (dissociation), $\text{s}^{-1}$

Association and dissociation constants

K_a = \frac{[PL]}{[P][L]} \qquad\qquad K_d = \frac{[P][L]}{[PL]} = \frac{1}{K_a}

$K_a$ : association constant, $\text{M}^{-1}$
$K_d$ : dissociation constant, $\text{M}$
$[P]$ , $[L]$ , $[PL]$ : free protein, free ligand, and complex concentrations, $\text{M}$ each

Fractional saturation

\Theta = \frac{r}{n}

$\Theta$ : fractional saturation, dimensionless
$r$ : average ligand molecules bound per protein molecule, dimensionless
$n$ : total binding sites per protein molecule, dimensionless

The Scatchard equation

Very important

\frac{\Theta}{[L]} = \frac{n}{K_d} - \frac{\Theta}{K_d}

$\Theta$ : fractional saturation, dimensionless
$[L]$ : free (unbound) ligand concentration, $\text{M}$
$n$ : total binding sites per protein molecule, dimensionless
$K_d$ : dissociation constant, $\text{M}$

The Scatchard equation

Very important

\frac{r}{[L]} = nK_a - rK_a = K_a(n - r)

$r$ : average ligand molecules bound per protein molecule, dimensionless
$[L]$ : free ligand concentration, $\text{M}$
$n$ : total binding sites per protein molecule, dimensionless
$K_a$ : association constant, $\text{M}^{-1}$

Full note →

Shear stress

Shear rate

Kinematic viscosity

Power law equation (Ostwald–de Waele model)

Apparent viscosity of a power-law fluid

Newton's law of viscosity

Bingham plastics

Casson plastics

Herschel–Bulkley model

Reynolds number for stirred tank

Reynolds number for straight pipe

Froude number for stirred tank

Power number (Newton number, NpN_pNp​)

Power in laminar flow

Impeller tip speed

Flow number (FlF_lFl​ / FQF_QFQ​)

If power is given

If power number is given

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)

Rate of heat transfer by conduction

Rate of heat transfer by convection

Nusselt number

Prandtl number

Conductive resistance of a wall

Conductive resistance of a wall

Rate of heat transfer using U

Overall heat transfer coefficient from film coefficients

Logarithmic mean temperature difference (LMTD)

Metabolic heat from oxygen consumption (Cooney's correlation)

Scale-up heat transfer challenge

Dilution rate, HRT, and washout

General equation of aerobic growth

Respiratory quotient (RQ)

Degree of reduction (γ\gammaγ)

Electron yields (η\etaη)

General equation with product formation

Yield coefficients

Theoretical oxygen demand (ThOD)

Elemental composition of biomass

Force–displacement form

Pressure–volume form

Internal energy

Internal energy

Enthalpy

Second law of thermodynamics

Entropy of the surroundings

Gibbs free energy

Ligand binding equilibrium

Association and dissociation constants

Fractional saturation

The Scatchard equation

The Scatchard equation

Power number (Newton number, $N_p$ )

Flow number ( $F_l$ / $F_Q$ )

$k_L$ under film theory

Gas hold-up ( $\varepsilon_g$ )

Gas hold-up ( $\varepsilon_g$ )

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

Degree of reduction ( $\gamma$ )

Electron yields ( $\eta$ )