This page collects every essential bioprocess engineering formula into a single reference. Whether you need a cell growth equation, a mass transfer formula, or a process economics calculation, you will find it here. Bookmark this formula cheat sheet—each equation links to the corresponding free calculator where you can plug in your own numbers.
1. Cell Growth Kinetics Formulas
μ = ln(X2 / X1) / (t2 − t1)
Specific growth rate (h&supmin;¹)
X = cell concentration (cells/mL or g/L)
t = time (h)
Specific growth rate (h&supmin;¹)
X = cell concentration (cells/mL or g/L)
t = time (h)
td = ln(2) / μ
Doubling time (h)
Doubling time (h)
X(t) = X0 × eμt
Exponential growth model
X0 = initial cell concentration
Exponential growth model
X0 = initial cell concentration
μ = μmax × S / (Ks + S)
Monod equation — substrate-limited growth
μmax = maximum specific growth rate (h&supmin;¹)
S = substrate concentration (g/L)
Ks = half-saturation constant (g/L)
Monod equation — substrate-limited growth
μmax = maximum specific growth rate (h&supmin;¹)
S = substrate concentration (g/L)
Ks = half-saturation constant (g/L)
IVC = ∫ X(t) dt ≈ Σ ½(Xi + Xi+1) × (ti+1 − ti)
Integral of viable cells (cells·day/mL) — trapezoidal rule
Integral of viable cells (cells·day/mL) — trapezoidal rule
qP = ΔP / ΔIVC
Cell-specific productivity (pg/cell/day or g/cell/h)
P = product concentration (g/L)
Cell-specific productivity (pg/cell/day or g/cell/h)
P = product concentration (g/L)
Track Growth in Real Time
Log cell counts, viability, and metabolites. Get automatic growth rate and IVC calculations.
CellTrack App →2. Mass Balance & Yield Equations
YX/S = ΔX / ΔS
Biomass yield on substrate (g biomass / g substrate)
Biomass yield on substrate (g biomass / g substrate)
YX/O = ΔX / ΔO2
Biomass yield on oxygen (g biomass / mol O2)
Biomass yield on oxygen (g biomass / mol O2)
Carbon balance:
Substratein = Biomass + CO2 + Product + Residual substrate
Substratein = Biomass + CO2 + Product + Residual substrate
RQ = CO2 produced / O2 consumed
Respiratory quotient (dimensionless)
RQ = 1.0 for glucose fully oxidized
RQ > 1.0 suggests overflow metabolism (e.g., ethanol, acetate)
Respiratory quotient (dimensionless)
RQ = 1.0 for glucose fully oxidized
RQ > 1.0 suggests overflow metabolism (e.g., ethanol, acetate)
3. Fed-Batch Feeding Formulas
Exponential feed:
F(t) = (μ × X0 × V0) / (YX/S × Sf) × eμt
F(t) = feed rate (L/h)
X0 = initial biomass concentration (g/L)
V0 = initial volume (L)
Sf = feed substrate concentration (g/L)
F(t) = (μ × X0 × V0) / (YX/S × Sf) × eμt
F(t) = feed rate (L/h)
X0 = initial biomass concentration (g/L)
V0 = initial volume (L)
Sf = feed substrate concentration (g/L)
Linear feed:
F(t) = F0 + k × t
F0 = initial feed rate (L/h)
k = ramp rate (L/h²)
F(t) = F0 + k × t
F0 = initial feed rate (L/h)
k = ramp rate (L/h²)
Substrate mass balance in fed-batch:
dS/dt = F × Sf / V − μ × X / YX/S − m × X
m = maintenance coefficient (g substrate / g biomass / h)
dS/dt = F × Sf / V − μ × X / YX/S − m × X
m = maintenance coefficient (g substrate / g biomass / h)
Calculate Your Feed Profile
Enter biomass, yield coefficients, and feed concentration to generate exponential and linear feed schedules.
Fed-Batch Calculator →4. Oxygen Transfer
OTR = kLa × (C* − CL)
Oxygen transfer rate (mg/L/h or mmol/L/h)
kLa = volumetric mass transfer coefficient (h&supmin;¹)
C* = saturated DO concentration (≈7 mg/L at 37°C in air)
CL = actual dissolved oxygen (mg/L)
Oxygen transfer rate (mg/L/h or mmol/L/h)
kLa = volumetric mass transfer coefficient (h&supmin;¹)
C* = saturated DO concentration (≈7 mg/L at 37°C in air)
CL = actual dissolved oxygen (mg/L)
Van't Riet correlation (coalescing media):
kLa = 0.026 × (P/V)0.4 × vs0.5
Van't Riet correlation (non-coalescing media):
kLa = 0.002 × (P/V)0.7 × vs0.2
P/V in W/m³, vs in m/s, kLa in s&supmin;¹
kLa = 0.026 × (P/V)0.4 × vs0.5
Van't Riet correlation (non-coalescing media):
kLa = 0.002 × (P/V)0.7 × vs0.2
P/V in W/m³, vs in m/s, kLa in s&supmin;¹
OUR = qO2 × X
Oxygen uptake rate (mmol/L/h)
qO2 = specific oxygen consumption rate (mmol/g/h)
Oxygen uptake rate (mmol/L/h)
qO2 = specific oxygen consumption rate (mmol/g/h)
Estimate kLa for Your Bioreactor
Enter vessel dimensions, impeller specs, and gas flow to calculate kLa and OTR.
OTR & kLa Estimator →5. Impeller & Mixing
P = Np × ρ × N³ × Di&sup5;
Ungassed impeller power draw (W)
Np = power number (dimensionless, e.g., 5.0 for Rushton)
ρ = fluid density (kg/m³)
N = impeller speed (rps)
Di = impeller diameter (m)
Ungassed impeller power draw (W)
Np = power number (dimensionless, e.g., 5.0 for Rushton)
ρ = fluid density (kg/m³)
N = impeller speed (rps)
Di = impeller diameter (m)
Tip speed:
vtip = π × N × Di
N in rps, Di in m, vtip in m/s
vtip = π × N × Di
N in rps, Di in m, vtip in m/s
Reynolds number:
Re = ρ × N × Di² / μ
μ = dynamic viscosity (Pa·s)
Re > 10,000 = fully turbulent (typical for aqueous fermentation)
Re = ρ × N × Di² / μ
μ = dynamic viscosity (Pa·s)
Re > 10,000 = fully turbulent (typical for aqueous fermentation)
Power per unit volume:
P/V = Ptotal / Vliquid
Typical ranges: 0.5–3 W/L (mammalian), 2–10 W/L (microbial)
P/V = Ptotal / Vliquid
Typical ranges: 0.5–3 W/L (mammalian), 2–10 W/L (microbial)
Mixing time (approximate):
tmix ∝ (V / P)1/3
tmix ∝ (V / P)1/3
Scale-Up Calculator
Compare constant P/V, tip speed, kLa, Re, and mixing time across scales.
Scale-Up Calculator →6. Heat Transfer Formulas
Qmet = OUR × V × 460 kJ/mol
Metabolic heat generation (W)
OUR in mol/L/s, V in L
460 kJ/mol is the heat of combustion per mol O2
Metabolic heat generation (W)
OUR in mol/L/s, V in L
460 kJ/mol is the heat of combustion per mol O2
Q = U × A × LMTD
Heat transfer rate (W)
U = overall heat transfer coefficient (W/m²/K)
A = heat transfer area (m²)
LMTD = log-mean temperature difference (K)
Heat transfer rate (W)
U = overall heat transfer coefficient (W/m²/K)
A = heat transfer area (m²)
LMTD = log-mean temperature difference (K)
LMTD = (ΔT1 − ΔT2) / ln(ΔT1 / ΔT2)
ΔT1 = Tbroth − Tcoolant,in
ΔT2 = Tbroth − Tcoolant,out
ΔT1 = Tbroth − Tcoolant,in
ΔT2 = Tbroth − Tcoolant,out
Coolant flow rate:
ṁcoolant = Q / (Cp × ΔT)
Cp = specific heat capacity (J/kg/K, water ≈ 4184)
ΔT = coolant temperature rise (K)
ṁcoolant = Q / (Cp × ΔT)
Cp = specific heat capacity (J/kg/K, water ≈ 4184)
ΔT = coolant temperature rise (K)
Heat Transfer Calculator
Calculate metabolic heat load and verify your cooling system can handle it.
Heat Transfer Calculator →7. Downstream Processing
DBC = mass bound / column volume
Dynamic binding capacity (mg/mL resin)
Measured at 10% breakthrough (DBC10%)
Dynamic binding capacity (mg/mL resin)
Measured at 10% breakthrough (DBC10%)
LRV = log10(load titer / filtrate titer)
Log reduction value — viral clearance
LRV ≥ 4 per step is typical regulatory expectation
Log reduction value — viral clearance
LRV ≥ 4 per step is typical regulatory expectation
Yield = (massout / massin) × 100%
Step yield (%)
Step yield (%)
Pool volume:
Vpool = CV × Vcolumn
CV = number of column volumes in the elution pool
Vpool = CV × Vcolumn
CV = number of column volumes in the elution pool
Related tools for applying these downstream formulas:
- Chromatography Calculator — Column sizing, binding capacity, and buffer volumes.
- Buffer Calculator — Buffer recipes and pH adjustment.
- Filtration / TFF Calculator — Membrane area, flux, and diafiltration volumes.
- Viral Clearance Calculator — LRV calculation and clearance study design.
- Protein A Resin Lifetime Calculator — Resin cycling and lifetime cost analysis.
8. Centrifugation
RCF = 1.118 × 10−5 × r × N²
Relative centrifugal force (× g)
r = radius (cm)
N = rotational speed (RPM)
Relative centrifugal force (× g)
r = radius (cm)
N = rotational speed (RPM)
Sigma factor (tubular bowl, simplified):
Σ = (2π × N² × L × r²) / g
L = bowl length (m)
r = bowl radius (m)
g = 9.81 m/s²
Σ = (2π × N² × L × r²) / g
L = bowl length (m)
r = bowl radius (m)
g = 9.81 m/s²
Scale-up by Q/Σ equivalence:
(Q / Σ)lab = (Q / Σ)production
Maintain constant Q/Σ to preserve separation performance
(Q / Σ)lab = (Q / Σ)production
Maintain constant Q/Σ to preserve separation performance
Centrifugation Scale-Up
Calculate sigma factors and scale between lab and production centrifuges.
Centrifugation Calculator →9. Process Economics Formulas
COGS/g = Total Cost / (Batches × Volume × Titer × Yield)
Cost of goods sold per gram of product
Total Cost = raw materials + labor + facility + overhead
Cost of goods sold per gram of product
Total Cost = raw materials + labor + facility + overhead
Productivity = Titer / Culture Duration
Volumetric productivity (g/L/day)
Volumetric productivity (g/L/day)
Annual output:
Output = Batches/year × V × Titer × DSP Yield
in kg/year or g/year
Output = Batches/year × V × Titer × DSP Yield
in kg/year or g/year
Model Your Process Economics
Build a full cost model for your upstream and downstream process.
Fermentation Economics →