Fed-Batch Accounting — For a fed-batch with no effluent, what expression gives the total mass of substrate utilized over a given time window?

Introduction:
Mass balances in fed-batch systems are central to calculating yields and specific rates. With no outflow, substrate accumulates in the reactor while being consumed by cells. Accounting for initial inventory, feed addition, and final leftover substrate yields the total substrate utilized.

Given Data / Assumptions:

• No liquid effluent (classic fed-batch).
• Known feed flow rate and feed substrate concentration over the time window.
• Measured initial and final substrate masses in the reactor.

Concept / Approach:

Overall substrate balance over the time window: utilized = (initial + fed in) − final. If feed conditions are constant, fed in = flow rate * substrate concentration in the feed * time. If variable, integrate over time. This expression ensures conservation and correctly handles changing volume due to feeding.

Step-by-Step Solution:

Verification / Alternative check:

Cross-check with biomass and product yields: Yx/s and Yp/s should produce consistent substrate utilization within measurement error.

Why Other Options Are Wrong:

A: Adds final instead of subtracting; overestimates consumption. C and D: Subtract feed term or omit units; inconsistent with conservation. E: Ignores initial and final inventories, causing bias.

Common Pitfalls:

Forgetting to account for changing volume and variable feed concentration; integrate when conditions vary.

Final Answer:

Initial mass of substrate in the reactor − final mass of substrate in the reactor + (flow rate * substrate concentration in the feed).