Exercise — Production of a vaccine molecule in a CSTR¶
Imagine a continuous bioreactor (CSTR) where a cell culture (Biomass X) grows using a substrate (S) and produces a vaccine molecule (P). The reactor has one inlet and one outlet, is perfectly mixed, has constant volume, and has temperature control.
Part 1 — General balances (mass + energy)¶
Write general mass and energy balances, identifying inlet/outlet and a generation/consumption term (reaction term), as shown in the previous lecture.
Hint:
Biomass
$$V\frac{dX}{dt} = F(X_{in}-X) + \text{(generation of X)}$$
Substrate
$$V\frac{dS}{dt} = F(S_{in}-S) + \text{(consumption or generation of S)}$$
Product
$$V\frac{dP}{dt} = F(P_{in}-P) + \text{(generation of P)}$$
Energy
$$\rho V C_p \frac{dT}{dt} = \dot Q + \text{(metabolic / reaction heat)} + \rho F C_p (T_{in}-T)$$
Part 2 — Process optimization to reach a specific steady-state concentration¶
You are working for a biopharmaceutical company in charge of producing a vaccine containing the molecule P from Part 1. Recent tests on the process have pointed out wash-out phenomena (flow rate too high) and substrate limitation. Your company asks you to redesign the protocol to produce at steady state 60 g/h of product P. Considering the balance from Part 1, choose optimized values for:
- operating temperature (try to stay realistic!),
- inlet flow rate $F$,
- feed substrate concentration $S_{in}$,
so that — according to your chosen reaction model — the reactor produces the desired concentration at steady state. How much can the inlet flow rate affect production (e.g., in terms of residence time $(V/F)$, production rate, product concentration, etc.)?
Part 3: Is your guess economically sustainable?¶
Your boss asks you to make a cost analysis on your new protocol. Assume that your company has a strict maximum process cost of €0.50 per gram of product (this includes substrate + electricity + cooling overhead).
Considering the following approximate cost data:
| Cost item | Typical realistic value (EU 2024-2025) | Notes |
|---|---|---|
| Substrate (glucose or similar carbon source) | € 3.5 – € 5.7 per kg | industrial bulk glucose price |
| Electricity (industrial tariff) | € 0.17 – € 0.19 per kWh | varies by EU state |
| Baseline power consumption (mixing + pumps) | 6 – 12 kW | assume average 8 kW |
| Additional cooling if T = 40°C | +25 – 50% more electricity | heat removal penalty |
| Cooling power (chiller equivalent load) | +4 – 8 kW | depends on metabolism/scale |
| Estimated hourly electricity cost (base 8 kW) | 8 kW × €0.18 ≈ €1.44/h | if T = 20–30°C |
| Estimated hourly electricity cost at 40°C | (~12 kW) × €0.18 ≈ €2.16/h | extra cooling included |
| Stainless-steel bioreactor CAPEX (1000 L) | € 100,000 – € 150,000 | purchase cost |
| Stainless-steel amortization | € 1.2 – € 1.8 per hour | assuming 8,000 h/year for 10 years |
| Amortization contribution per gram (20 g/h) | € 0.06 – € 0.09 per gram | divide amortization by g/h |
| Single-use bag reactor (1000 L, per campaign) | € 4,000 – € 8,000 per run | disposable + consumables |
| Single-use cost per hour (for 7-day run) | € 24 – € 48 per hour | divide cost by runtime |
| Single-use cost per gram (20 g/h) | € 1.2 – € 2.4 per g | much higher than steel if long run |
Estimate the approximate cost per gram of product for your chosen set of parameters. Is your guess economically sustainable?