In a plug flow reactor (PFR) used for bioprocessing, where along the reactor do cells encounter the highest substrate concentration, given that fresh feed enters at one end and flows without back-mixing?

Introduction / Context:
A plug flow reactor (PFR) is an idealized flow model widely applied in biochemical engineering to describe tubular bioreactors with negligible axial mixing. Understanding how substrate concentration changes along the reactor length is essential for predicting growth, product formation, and potential inhibition patterns for cells immobilized or suspended within the flow.

Given Data / Assumptions:

• Ideal plug flow: negligible axial back-mixing; radial mixing is fast.
• Fresh feed with the highest substrate concentration enters at the inlet.
• Substrate is consumed monotonically along the flow direction.
• No significant radial gradients after initial entrance effects (well-mixed cross-section).

Concept / Approach:
In a PFR, each fluid element advances down the reactor while reacting. Substrate concentration S decreases with axial coordinate z due to cellular uptake. Therefore, the maximum S occurs at z ≈ 0 (feed entrance), and S is lowest near the outlet. Cells exposed early in the reactor face the strongest driving force for uptake and, if relevant, the highest risk of substrate inhibition.

Step-by-Step Solution:
Identify model: plug flow implies S = S(z) with dS/dz < 0 for consuming reactions.At z = 0 (entrance), S = S_in (the highest concentration).As z increases, cells consume substrate, so S decreases along the length.Hence, cells located at the entrance experience the highest S.

Verification / Alternative check:
Mass balance for a first-order uptake: dS/dz = -k * S / u (u = superficial velocity) integrates to S(z) = S_in * exp(-k z / u), confirming a monotonic decline from the inlet to the outlet.

Why Other Options Are Wrong:
Near the effluent: S is lowest there after consumption.Midway: S has already dropped from inlet value.Near the wall: in an ideal PFR, radial gradients are negligible; wall proximity does not define the axial maximum.

Common Pitfalls:
Confusing PFR with CSTR behavior, assuming recirculation leads to uniform S, or overemphasizing wall effects in an ideal radial-mixing assumption.

Final Answer:
Cells located near the feed entrance