GATE Bioprocess Engineering formula sheet

Constant-volume batch. $k$ does not depend on concentration; it varies with temperature. Its units are $(\text{mol/L}{)}^{1-n}{\text{time}}^{-1}$.

Constant volume. $C_{A0}$ is the initial concentration of reactant A.

The fraction of fed A that has reacted. The concentration form holds only at constant density (liquid-phase systems).

The time needed to process one reactor volume of feed, measured at feed conditions (h). The space velocity $s$ is its reciprocal (h${}^{-1}$).

$z$ = signed charge number of the substrate, $\psi$ = signed electrostatic potential of the support (V), $F=96485$ C/mol, $R_g=8.314$ J mol${}^{-1}$ K${}^{-1}$, $T$ in K. Assumes a uniform potential in the enzyme's microenvironment and Boltzmann partitioning; the effect is largest at low ionic strength, because added salt screens the charge. Same signs ($z\psi >0$): substrate repelled, $K_{m,app}>K_m$. Opposite signs: substrate attracted, $K_{m,app}<K_m$. Partitioning alone leaves $V_{max}$ unchanged. More generally, any partition coefficient $K_p=S_{local}/S_b$ (electrostatic or hydrophobic) gives $K_{m,app}=K_m/K_p$ with $V_{max}$ unchanged.

Steady state; enzyme on the external surface only, no internal diffusion. Solve for $S_s$; the observed rate per unit area is either side of the equation.

Definition, valid in any phase; units ${\mathrm{h}}^{-1}$. $X$ is the biomass concentration (g/L). When death is negligible this is the growth rate itself; when death or endogenous decay is significant, the measured $(1/X)dX/dt$ is the net rate $\mu -k_d$ (see below). The number-based rate ${\mu }_N$ uses the cell number concentration $N$ instead of $X$.

Applies to the exponential phase only: constant $\mu$, no lag inside the interval, death neglected ($X=X_0$ at $t=t_0$). If death is significant, use the net rate $\mu -k_d$ in place of $\mu$. Use the natural log. A plot of $\ln X$ against $t$ is a straight line of slope $\mu$.

${\mu }_m$ = maximum specific growth rate (h${}^{-1}$); $K_s$ = saturation constant, the substrate concentration at which $\mu ={\mu }_m/2$ (g/L). For $S\gg K_s$, $\mu \approx {\mu }_m$. Every result below that contains $K_s$ assumes this form, with a single growth-limiting substrate.

Exponential phase of a batch culture, where $S\gg K_s$ so that $\mu \approx {\mu }_m$; no lag and no death.

$X$ and $S$ in g/L, $Y_{X/S}$ in g dry cells per g substrate. For a limiting substrate used up completely, $S_f\approx 0$. Maintenance and product formation lower the observed yield, so use the yield measured under the same conditions.

$m_{\text{nutrient}}$ in g/L; $w_E$ = mass fraction of element E in dry biomass, $\Delta X$ = biomass to be formed (g/L), $f_E$ = mass fraction of E in the nutrient compound (for N in $(\mathrm{N}{\mathrm{H}}_4{)}_2\mathrm{S}{\mathrm{O}}_4$, $f_N=28/132$). This is a minimum: it assumes the compound is the only source of E and that all of it is taken up.

In U/mg, where 1 U (enzyme unit) is the amount of enzyme that converts 1 μmol of substrate per minute under the stated assay conditions. It rises as contaminating protein is removed and is the working measure of purity for an enzyme or other active protein.

Yields use TOTAL activity (or total product mass), never specific activity. The overall yield is the product of the step yields.

$E^0$ is a constant of the electrode pair, fixed by calibration with buffers. $R=8.314$ J mol${}^{-1}$ K${}^{-1}$, $F=96485$ C mol${}^{-1}$, $T$ in K. The slope is proportional to absolute temperature.

SI units: $Q$ in m${}^3$ s${}^{-1}$, $\Delta P$ in Pa, $\rho$ (fluid density) in kg m${}^{-3}$. $A_0$ and $d_0$ are the orifice (or venturi throat) area and diameter, $D$ is the pipe diameter and $C_d$ the discharge coefficient. Because $Q\propto \sqrt{\Delta P}$, a 10 : 1 flow range needs a 100 : 1 pressure-drop range.