In fed-batch bioreactor modeling under standard assumptions (no outlet during feed, constant density), the time rate of change of reactor volume dV/dt is equal to which quantity?

Introduction / Context:
Fed-batch operation increases reactor volume over time by adding feed without removing broth (until harvest). A basic dynamic mass balance on the liquid reveals how volume evolves and links directly to concentration dynamics of substrates and biomass.

Given Data / Assumptions:

• No outlet flow during the feeding phase (closed to effluent).
• Liquid density is approximately constant; evaporation and entrainment are negligible.
• Reactor is well mixed.

Concept / Approach:
The unsteady overall volume balance is dV/dt = F_in − F_out. In fed-batch with no intentional effluent, F_out ≈ 0, so dV/dt = F_in. This identity underpins canonical fed-batch equations for state variables, e.g., d(XV)/dt = μXV − D_loss terms, and for concentrations via dilution effects Ċ = (generation − consumption)/V − (F_in/V) * C + (F_in/V) * C_feed when relevant.

Step-by-Step Solution:
Write volume balance: dV/dt = F_in − F_out.Apply fed-batch condition: F_out ≈ 0 during feed.Therefore, dV/dt = F_in (the feed rate).Consequences: dilution term (F_in/V) appears in all concentration balances.

Verification / Alternative check:
Integrating dV/dt = F_in over time yields V(t) = V0 + ∫ F_in dt, matching practical observations where liquid level rises by the cumulative feed volume.

Why Other Options Are Wrong:
Initial volume: a constant; does not represent a rate.Volume of solids: typically small and does not define the liquid volume rate.None of these: incorrect because the inflow rate is the correct equality.

Common Pitfalls:
Forgetting evaporation or sampling losses in real systems; assuming density changes invalidate the balance (they affect mass, not the volumetric identity under constant density assumption).

Final Answer:
The inlet volumetric flow rate (feed rate)