Design for a reversible endothermic liquid-phase reaction in a plug flow reactor (PFR): To minimize the required reactor volume, how should the temperature vary along the reactor length?

Introduction / Context:
Optimal temperature policy in tubular reactors strongly affects conversion and required volume, especially for reversible reactions where equilibrium plays a central role. For endothermic systems, higher temperature tends to favor products at equilibrium and increases intrinsic kinetics.

Given Data / Assumptions:

• Reversible, endothermic reaction in liquid phase.
• Plug flow reactor behavior (no axial mixing).
• Temperature is a manipulable variable subject to an upper limit.

Concept / Approach:
For endothermic reactions: (i) the forward rate constant increases with temperature; (ii) the equilibrium conversion increases with temperature. Thus, operating as hot as permitted simultaneously enhances kinetics and shifts equilibrium toward products, reducing required volume for a target conversion.

Step-by-Step Solution:
Endothermic reaction: higher T raises both k and equilibrium conversion Xeq(T).In a PFR, local rate r(T, C) determines dX/dV; higher r reduces V for a given ΔX.Therefore, the optimal policy is to keep T at the maximum allowable value along the entire length, subject to constraints (materials, safety).

Verification / Alternative check:
Variational analysis and heuristic design rules both indicate “heat it up” for endothermic reversible systems to push equilibrium and kinetics favorably. Opposite guidance applies to reversible exothermic systems, where lower temperatures can favor equilibrium conversion.

Why Other Options Are Wrong:

• Decreasing or low T: reduces both rate and equilibrium conversion, increasing required V.
• Increasing T with length: better than decreasing, but not as effective as staying at the highest allowable throughout.
• Non-monotonic profiles: no advantage over uniformly maximal T for endothermic reversible systems.

Common Pitfalls:
Confusing strategies for exothermic vs. endothermic reversible reactions; they are opposite in terms of best equilibrium leverage.

Final Answer:
Maintained at the highest allowable temperature throughout