Fed-Batch Exponential Feeding Rate Calculator
In an exponentially fed fed-batch culture the substrate feed rises with time so that the specific growth rate μ stays constant. This calculator gives the feed rate F(t) needed at a time t after the feed started, in L/h and in mL/h, together with the biomass in the vessel X·V in g.
The feed follows from the biomass and substrate balances of a substrate-limited fed-batch. The biomass grows as X·V = X0·V0·exp(μ·t), the specific substrate use is q = μ / Yxs, and the feed must supply F·Sf / V = q·X. Solving for F gives:
F(t) = (μ / Yxs) × X0 × V0 × exp(μ × t) / Sf [L/h]
The calculation assumes that the substrate is the limiting nutrient, that residual substrate and maintenance are neglected, and that μ stays constant. If μ × t is above 700, or the result is not a finite number, the calculator shows an out of range error instead of a value.
Results
How to use the calculator
- Enter the specific growth rate μ you want to hold constant, in 1/h.
- Enter the biomass yield on substrate Yxs, in g/g.
- Enter the initial biomass concentration X0 in g/L and the initial culture volume V0 in L, both taken when the feed starts.
- Enter the substrate concentration of the feed Sf, in g/L.
- Enter the time since the feed started t, in h. A value of 0 gives the starting feed rate.
- Read the feed rate F in L/h and mL/h, the biomass in the vessel X·V in g, and the formula line with your values. Results update as you type; the Calculate button runs the same calculation.
- Set the feed pump to the rate for the current time and repeat for later times, since the required feed rises exponentially.
Frequently asked questions
What does this calculator compute?
It computes the substrate feed rate F(t) = (μ / Yxs) × X0 × V0 × exp(μ × t) / Sf, in L/h and mL/h, that keeps the specific growth rate μ constant in an exponentially fed fed-batch culture, together with the biomass in the vessel X × V = X0 × V0 × exp(μ × t), in g.
Where does the feeding equation come from?
It follows from the biomass and substrate balances of a substrate-limited fed-batch. The biomass grows as X × V = X0 × V0 × exp(μ × t), the specific substrate use is q = μ / Yxs, and the feed must supply F × Sf / V = q × X. Solving for F gives F = q × X × V / Sf.
What assumptions does the calculation make?
The substrate is the limiting nutrient, residual substrate and maintenance are neglected, and the specific growth rate μ stays constant over the whole feeding time.
What happens at t = 0, and when is a result out of range?
At t = 0, exp(μ × t) equals 1, so the result is the starting feed rate (μ / Yxs) × X0 × V0 / Sf. If μ × t is above 700, or the result is not a finite number, the calculator shows an out of range error instead of a value.
Disclaimer: this calculator gives an estimate only and is not a substitute for a qualified engineer. The feed rate follows from the biomass and substrate balances of a substrate-limited fed-batch culture (source: IIT Guwahati, BTech protocol Experiment 9, Fed-batch operation to maintain exponential growth phase, equations 1 to 4 and 9: biomass X·V = X0·V0·exp(μ·t), substrate use q = μ / Yxs, feed F·Sf / V = q·X). Keeping the culture on this exponential profile needs control of the feed pump.