Upstream formulas

$X$ 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.

Mean response over all runs at the high level minus the mean over all runs at the low level. A positive effect means that raising the factor raises the response within the range tested. An interaction effect $E_{ij}$ is found the same way, grouping the runs by the sign of the product $x_i{x}_j$. In a regression on coded variables the coefficient is half the effect, ${\beta }_i=E_i/2$, because $x_i$ changes by 2 between the levels.

Number of coefficients $=\frac{(k+1)(k+2)}{2}$ (10 for $k=3$). Fitted by least squares; the stationary point comes from setting every $\partial Y/\partial x_i=0$, and it may be a maximum, a minimum or a saddle point, so its nature must be checked before it is called the optimum.

$N$ = number of viable organisms (use the total number in the vessel for design, not the concentration); $k_d$ = specific death rate constant in min${}^{-1}$ or s${}^{-1}$. Holds at constant temperature.

Time at a fixed temperature for a 90% kill (one log cycle). $D$ and $k_d$ must refer to the same temperature.

$T$ in kelvin, $R=8.314$ J mol${}^{-1}$ K${}^{-1}$, $E_d$ = activation energy for death in J mol${}^{-1}$, $A$ has the units of $k_d$.

$z$ = temperature rise that reduces $D$ tenfold, assuming ${\log }_{10}D$ falls linearly with temperature. It is a temperature difference, so its value is the same in °C and K.

Dimensionless, natural log. $N_0=n_{0}V$ (spore concentration × medium volume), $N$ = acceptable number of survivors per batch (for example $10^{-3}$).

${\nabla }_{\text{heat}}$ and ${\nabla }_{\text{cool}}$ are the integrals of $k_d$ over the heating and cooling temperature–time profiles; the holding period at constant temperature supplies the rest.

$\bar{u}$ = mean velocity, $Q$ = volumetric flow rate, $d$ = tube internal diameter, $L$ = holding-tube length, $k_d$ at the holding temperature. Valid for a flat (turbulent) velocity profile, negligible axial dispersion and near-instant heating and cooling; otherwise add ${\nabla }_{\text{heat}}$ and ${\nabla }_{\text{cool}}$.

$K$ = filter constant (per unit depth), which depends on air velocity, fibre diameter and packing; $L_{90}$ = bed depth that removes 90%, so ${\log }_{10}(N_0/N)=L/L_{90}$. Collection efficiency ${\eta }_{\text{filter}}=1-N/N_0$, so $\ln \frac{1}{1-{\eta }_{\text{filter}}}=KL$.

$n_0$ = organisms per m${}^3$ of air, $Q$ = air flow rate (m${}^3$ per unit time), $t$ = duration of the fermentation, $N$ = acceptable number of organisms passing in that time (for example $10^{-3}$).