Formula: \( \mu = \frac{\ln N_2 - \ln N_1}{t_2 - t_1} \)
Description: Calculates the specific (exponential) growth rate of a microbial population between two time points.
Example: If N1 = 1×10^3 CFU/mL at t1 = 0 h and N2 = 1×10^6 CFU/mL at t2 = 4 h, μ = (ln(1e6)-ln(1e3))/4 ≈ 1.723 h⁻¹.
Formula: \( t_d = \frac{\ln 2}{\mu} \)
Description: Calculates the time required for a microbial population to double in size during exponential growth.
Example: If μ = 0.5 h⁻¹, td = ln(2)/0.5 ≈ 1.386 h.
Formula: \( g = \frac{\ln 2}{\mu} \)
Description: Time required for one generation (doubling) of microbial cells, equivalent to doubling time.
Example: If μ = 0.693 h⁻¹, g = ln(2)/0.693 ≈ 1 h.
Formula: \( N = N_0 \cdot e^{\mu t} \)
Description: Calculates the population size at time t during exponential growth.
Example: If N0 = 1×10^3 CFU/mL, μ = 0.5 h⁻¹, t = 2 h, N = 1e3 * e^(0.5*2) ≈ 2.718×10^3 CFU/mL.
Formula: \( n = \frac{\ln N - \ln N_0}{\ln 2} \)
Description: Calculates the number of microbial generations between two population sizes.
Example: If N0 = 1×10^3 CFU/mL and N = 1×10^6 CFU/mL, n = (ln(1e6)-ln(1e3))/ln(2) ≈ 9.966 generations.
Formula: \( Y = \frac{X - X_0}{S} \)
Description: Calculates the biomass yield per unit of substrate consumed.
Example: If X0 = 0.1 g/L, X = 2.1 g/L, S = 5 g/L, Y = (2.1-0.1)/5 = 0.4 g biomass/g substrate.
Formula: \( \mu = \frac{\mu_{\max} \cdot S}{K_s + S} \)
Description: Describes the relationship between specific growth rate and substrate concentration.
Example: If μ_max = 0.8 h⁻¹, S = 2 g/L, Ks = 1 g/L, μ = 0.8 * 2 / (1 + 2) ≈ 0.533 h⁻¹.
Formula: \( t = \frac{\ln X - \ln X_0}{\mu} \)
Description: Calculates the time required to reach a specific biomass concentration in batch culture.
Example: If X0 = 0.1 g/L, X = 10 g/L, μ = 0.5 h⁻¹, t = (ln(10)-ln(0.1))/0.5 ≈ 9.21 h.
Formula: \( \mu = \frac{\mu_{\max} \cdot S}{K_s + S + \frac{S^2}{K_i}} \)
Description: Describes growth rate with substrate inhibition.
Example: If μ_max = 0.8 h⁻¹, S = 2 g/L, Ks = 1 g/L, Ki = 10 g/L, μ = (0.8 * 2)/(1 + 2 + (2^2/10)) ≈ 0.516 h⁻¹.
Formula: \( P_x = \frac{X - X_0}{t} \)
Description: Calculates the biomass productivity in a culture.
Example: If X0 = 0.5 g/L, X = 5 g/L, t = 10 h, Px = (5-0.5)/10 = 0.45 g/L/h.
Formula: \( q_x = \mu \cdot X \)
Description: Calculates the specific biomass production rate in a culture.
Example: If μ = 0.4 h⁻¹, X = 3 g/L, qx = 0.4 * 3 = 1.2 g/L/h.
Formula: \( K_i = \frac{S^2}{\left( \frac{\mu_{\max}}{\mu} - 1 \right) \cdot K_s - S} \)
Description: Calculates the substrate inhibition constant in the Monod-Haldane model.
Example: If S = 2 g/L, μ_max = 0.8 h⁻¹, μ = 0.4 h⁻¹, Ks = 1 g/L, Ki = 2^2/((0.8/0.4-1)*1-2) = 4/(-1) = -4 g/L (indicating model limitations).
Formula: \( q_s = \frac{\mu}{Y} \)
Description: Calculates the specific substrate consumption rate.
Example: If μ = 0.5 h⁻¹, Y = 0.4 g biomass/g substrate, qs = 0.5/0.4 = 1.25 g substrate/g biomass/h.
Formula: \( m = q_s - \frac{\mu}{Y_{\max}} \)
Description: Calculates the maintenance energy coefficient for substrate consumption.
Example: If qs = 1.25 g substrate/g/h, μ = 0.5 h⁻¹, Y_max = 0.5 g biomass/g substrate, m = 1.25 - (0.5/0.5) = 0.25 g substrate/g/h.
Formula: \( r_s = \frac{\mu \cdot X}{Y} \)
Description: Calculates the rate of substrate utilization in a microbial culture.
Example: If μ = 0.5 h⁻¹, X = 2 g/L, Y = 0.4 g biomass/g substrate, r_s = (0.5 * 2)/0.4 = 2.5 g/L/h.
Formula: \( q_s = \frac{S_0 - S}{X \cdot t} \)
Description: Calculates the specific rate of substrate uptake by biomass.
Example: If S0 = 10 g/L, S = 2 g/L, X = 2 g/L, t = 4 h, qs = (10-2)/(2*4) = 1 g substrate/g biomass/h.
Formula: \( \eta = \frac{S_0 - S}{S_0} \cdot 100 \)
Description: Calculates the percentage of substrate converted by the microbial culture.
Example: If S0 = 10 g/L, S = 3 g/L, η = (10-3)/10 * 100 = 70%.
Formula: \( Y_m = \frac{\mu}{q_s - m} \)
Description: Calculates the yield coefficient accounting for maintenance energy.
Example: If μ = 0.5 h⁻¹, qs = 1.25 g substrate/g/h, m = 0.25 g substrate/g/h, Ym = 0.5/(1.25-0.25) = 0.5 g biomass/g substrate.
Formula: \( q_{\text{glu}} = \frac{\text{Glu}_0 - \text{Glu}}{X \cdot t} \)
Description: Calculates the specific glucose uptake rate by biomass.
Example: If Glu0 = 20 g/L, Glu = 5 g/L, X = 3 g/L, t = 5 h, q_glu = (20-5)/(3*5) = 1 g glucose/g biomass/h.
Formula: \( D = \frac{F}{V} \)
Description: Calculates the dilution rate in a continuous culture system.
Example: If F = 0.2 L/h and V = 1 L, D = 0.2/1 = 0.2 h⁻¹.
Formula: \( X = Y \cdot (S_0 - S) \)
Description: Calculates the steady-state biomass concentration in a chemostat.
Example: If Y = 0.5 g biomass/g substrate, S0 = 10 g/L, S = 2 g/L, X = 0.5 * (10-2) = 4 g/L.
Formula: \( D_{\text{washout}} = \frac{\mu_{\max} \cdot S}{K_s + S} \)
Description: Calculates the dilution rate at which biomass is washed out in a chemostat.
Example: If μ_max = 0.8 h⁻¹, S = 2 g/L, Ks = 1 g/L, D_washout = 0.8 * 2 / (1 + 2) ≈ 0.533 h⁻¹.
Formula: \( P_x = D \cdot X \)
Description: Calculates the biomass productivity in a chemostat.
Example: If D = 0.2 h⁻¹, X = 5 g/L, Px = 0.2 * 5 = 1 g/L/h.
Formula: \( \text{OUR} = q_{O_2} \cdot X \)
Description: Calculates the oxygen uptake rate of a microbial culture.
Example: If qO2 = 2 mmol O₂/g/h, X = 3 g/L, OUR = 2 * 3 = 6 mmol O₂/L/h.
Formula: \( q_{O_2} = \frac{\text{OUR}}{X} \)
Description: Calculates the specific oxygen uptake rate per unit biomass.
Example: If OUR = 10 mmol O₂/L/h, X = 2 g/L, qO2 = 10/2 = 5 mmol O₂/g/h.
Formula: \( k_L a = \frac{\text{OUR}}{C^* - C} \)
Description: Calculates the oxygen transfer coefficient in a bioreactor.
Example: If OUR = 5 mmol O₂/L/h, C* = 0.2 mmol O₂/L, C = 0.05 mmol O₂/L, kLa = 5/(0.2-0.05) ≈ 33.33 h⁻¹.
Formula: \( \text{CER} = q_{CO_2} \cdot X \)
Description: Calculates the carbon dioxide evolution rate in a microbial culture.
Example: If qCO2 = 1.5 mmol CO₂/g/h, X = 4 g/L, CER = 1.5 * 4 = 6 mmol CO₂/L/h.
Formula: \( \text{RQ} = \frac{\text{CER}}{\text{OUR}} \)
Description: Calculates the respiratory quotient, the ratio of CO₂ produced to O₂ consumed.
Example: If CER = 6 mmol CO₂/L/h, OUR = 5 mmol O₂/L/h, RQ = 6/5 = 1.2.
Formula: \( Y_{x/O_2} = \frac{X}{O_2} \)
Description: Calculates the biomass yield per unit of oxygen consumed.
Example: If X = 2 g/L, O2 = 10 mmol O₂/L, Yx/O2 = 2/10 = 0.2 g biomass/mmol O₂.
Formula: \( \text{OTR} = k_L a \cdot (C^* - C) \)
Description: Calculates the oxygen transfer rate in a bioreactor.
Example: If kLa = 50 h⁻¹, C* = 0.2 mmol O₂/L, C = 0.05 mmol O₂/L, OTR = 50 * (0.2-0.05) = 7.5 mmol O₂/L/h.
Formula: \( C_{\text{crit}} = \frac{\text{OUR}}{k_L a} \)
Description: Calculates the critical oxygen concentration where oxygen becomes limiting.
Example: If OUR = 5 mmol O₂/L/h, kLa = 50 h⁻¹, Ccrit = 5/50 = 0.1 mmol O₂/L.
Formula: \( q_p = \frac{P}{X \cdot t} \)
Description: Calculates the specific product formation rate.
Example: If P = 10 g/L, X = 2 g/L, t = 5 h, qp = 10/(2*5) = 1 g product/g biomass/h.
Formula: \( Q = \frac{P}{t} \)
Description: Calculates the volumetric productivity of a product in a bioreactor.
Example: If P = 15 g/L, t = 10 h, Q = 15/10 = 1.5 g/L/h.
Formula: \( Y_{p/s} = \frac{P}{S} \)
Description: Calculates the product yield per unit of substrate consumed.
Example: If P = 8 g/L, S = 20 g/L, Yp/s = 8/20 = 0.4 g product/g substrate.
Formula: \( q_p = \mu \cdot Y_{p/x} \)
Description: Calculates the specific productivity based on growth rate and product yield per biomass.
Example: If μ = 0.4 h⁻¹, Yp/x = 0.5 g product/g biomass, qp = 0.4 * 0.5 = 0.2 g product/g biomass/h.
Formula: \( Y_{p/x} = \frac{P}{X} \)
Description: Calculates the product yield per unit of biomass produced.
Example: If P = 5 g/L, X = 2 g/L, Yp/x = 5/2 = 2.5 g product/g biomass.
Formula: \( q_p = \frac{P_0 - P}{X \cdot t} \)
Description: Calculates the specific rate of product uptake or degradation by biomass.
Example: If P0 = 10 g/L, P = 4 g/L, X = 2 g/L, t = 3 h, qp = (10-4)/(2*3) = 1 g product/g biomass/h.
Formula: \( Q = \frac{P}{V \cdot t} \)
Description: Calculates the product productivity in a fed-batch culture.
Example: If P = 50 g, V = 10 L, t = 20 h, Q = 50/(10*20) = 0.25 g/L/h.
Formula: \( X = \text{OD} \cdot k \)
Description: Converts optical density (OD) to cell concentration using a calibration factor.
Example: If OD = 0.5, k = 1×10^8 cells/mL/OD, X = 0.5 * 1e8 = 5×10^7 cells/mL.
Formula: \( \text{CFU/mL} = \frac{\text{Colonies} \cdot \text{Dilution Factor}}{\text{Volume Plated}} \)
Description: Calculates the number of viable cells per mL using plate count method.
Example: If 50 colonies, dilution factor = 10^5, volume plated = 0.1 mL, CFU/mL = (50 * 10^5)/0.1 = 5×10^7 CFU/mL.
Formula: \( \text{Log Reduction} = \log_{10} \left( \frac{N_0}{N} \right) \)
Description: Calculates the logarithmic reduction in microbial population due to disinfection.
Example: If N0 = 1×10^6 CFU/mL, N = 1×10^2 CFU/mL, Log Reduction = log10(1e6/1e2) = 4.
Formula: \( D = \frac{t}{\log_{10} \left( \frac{N_0}{N} \right)} \)
Description: Calculates the time required to reduce microbial population by 90% at a specific condition.
Example: If t = 2 min, N0 = 1×10^6 CFU/mL, N = 1×10^5 CFU/mL, D = 2/log10(1e6/1e5) = 2/1 = 2 min.
Formula: \( Z = \frac{T_2 - T_1}{\log_{10} \left( \frac{D_1}{D_2} \right)} \)
Description: Calculates the temperature change required to change the D-value by a factor of 10.
Example: If T1 = 60°C, T2 = 70°C, D1 = 10 min, D2 = 1 min, Z = (70-60)/log10(10/1) = 10/1 = 10°C.
Formula: \( F = D \cdot \log_{10} \left( \frac{N_0}{N} \right) \)
Description: Calculates the equivalent time at a reference temperature to achieve a specific microbial reduction.
Example: If D = 2 min, N0 = 1×10^6 CFU/mL, N = 1×10^2 CFU/mL, F = 2 * log10(1e6/1e2) = 2 * 4 = 8 min.
Formula: \( F_0 = t \cdot 10^{\frac{T - 121}{Z}} \)
Description: Calculates the equivalent sterilization time at 121°C based on temperature and Z-value.
Example: If t = 15 min, T = 115°C, Z = 10°C, F0 = 15 * 10^((115-121)/10) = 15 * 10^(-0.6) ≈ 3.57 min.
Formula: \( k_d = \frac{\ln N_0 - \ln N}{t} \)
Description: Calculates the specific death rate of a microbial population.
Example: If N0 = 1×10^6 CFU/mL, N = 1×10^4 CFU/mL, t = 2 h, kd = (ln(1e6)-ln(1e4))/2 ≈ 2.303 h⁻¹.
Formula: \( k_d = A \cdot e^{-\frac{E_a}{R \cdot T}} \)
Description: Describes the temperature dependence of the microbial death rate.
Example: If A = 1×10^10 h⁻¹, Ea = 50 kJ/mol, T = 333 K, kd = 1e10 * e^(-50000/(8.314*333)) ≈ 0.0014 h⁻¹.
Formula: \( Y_{x/N} = \frac{X}{N} \)
Description: Calculates the biomass yield per unit of nitrogen consumed.
Example: If X = 4 g/L, N = 0.5 g/L, Yx/N = 4/0.5 = 8 g biomass/g nitrogen.
Formula: \( q_N = \frac{N_0 - N}{X \cdot t} \)
Description: Calculates the specific nitrogen uptake rate by biomass.
Example: If N0 = 1 g/L, N = 0.2 g/L, X = 2 g/L, t = 4 h, qN = (1-0.2)/(2*4) = 0.1 g nitrogen/g biomass/h.