Engineering
Monod Bioreactor Microbial Growth, Doubling Time & Chemostat Washout
Explore Monod growth, intrinsic doubling time and the washout boundary of an ideal single-substrate chemostat without decay or biomass retention.
Distinguish microbial growth from net biomass accumulation
The Monod relation is an empirical saturation model for a limiting substrate. This example uses mu_max in h^-1 and Ks, S and feed Sin in g/L. Ks is the half-saturation concentration within this model, not an exact universal nutrient constant. Parameters depend on the organism and conditions.
LN(2)/mu is an intrinsic exponential-growth doubling time at constant substrate without losses. In a continuous reactor, washout also removes biomass; at a nonzero steady state mu=D and biomass concentration does not double.
The washout boundary mu(Sin) assumes an ideal well-mixed vessel, sterile feed, one limiting substrate, constant volume and parameters, no endogenous decay, no inhibition and no cell retention. It does not guarantee safe operation of an actual fermentation or wastewater process. Above the boundary biomass tends toward washout over time, not instant disappearance.
B4 is the local substrate concentration for the growth example; B5 is the separate feed concentration used for the boundary. Zero substrate produces zero growth and no finite intrinsic doubling time. Ks and mu_max must be positive.
| Bioprocess Kinetic Metric | Formula | mu_max=0.60/h, Ks=4 g/L, S=12 g/L, Sin=20 g/L | At S = Ks = 4 g/L |
|---|---|---|---|
| Specific Growth Rate mu | (B2 * B4) / (B3 + B4) | 7.2 / 16 = 0.4500 hr^-1 | 0.3000 hr^-1 (0.5 * mu_max!) |
| Cell Doubling Time t_d (hr) | LN(2) / ((B2 * B4) / (B3 + B4)) | 0.6931 / 0.45 = 1.54 hours | 2.31 hours |
| Critical Washout Rate D_crit | (B2 * B5) / (B3 + B5) | 12 / 24 = 0.5000 hr^-1 ceiling | 0.5000 hr^-1 |
Monod Specific Microbial Growth Rate mu = (mu_max * S) / (K_s + S)
Evaluates specific growth rate (hr^-1 or day^-1) from mu_max (B2), half-saturation K_s (B3), and substrate S (B4).
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<0),"Invalid inputs",ROUND(B2*B4/(B3+B4),4)))Intrinsic doubling time at constant substrate
Use the unrounded growth rate and return hours. This is not net doubling time in a flowing chemostat.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<0),"Invalid inputs",IF(B4=0,"No finite doubling time",ROUND(LN(2)/(B2*B4/(B3+B4)),2))))Ideal chemostat washout boundary
Evaluate growth at the feed concentration Sin, not the current vessel substrate S. This is an ideal model boundary rather than a recommended operating setpoint.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<0),"Invalid inputs",ROUND(B2*B5/(B3+B5),4)))Try the example
Use the original values or change them to test your understanding.