Distributed versus lumped parameter systems: which process behaves as a distributed parameter system with spatial gradients in state variables?

Introduction / Context:
Systems are modeled as lumped or distributed depending on whether state variables (temperature, concentration, pressure) can be assumed uniform within control volumes. Recognizing distributed systems helps select the correct mathematical tools (partial differential equations) and control strategies.

Given Data / Assumptions:

• Tubular reactor: plug flow behavior with axial (and possibly radial) gradients.
• CSTR: ideally well-mixed; state variables uniform within the tank.
• On–off controller: a control law, not a physical distributed system.

Concept / Approach:
Distributed parameter systems require PDEs because variables depend on both time and space (e.g., T(z,t), C_A(z,t)). A tubular reactor develops concentration and temperature profiles along its length; thus it is inherently distributed. By contrast, a CSTR assumes complete mixing so ordinary differential equations suffice, making it lumped.

Step-by-Step Solution:
Identify the presence of spatial gradients: tubular reactor has axial profiles.Classify according to modeling framework: PDEs → distributed.Select tubular reactor as the correct example.

Verification / Alternative check:
Introductory reaction engineering texts derive the PFR material and energy balances as PDEs (or steady ODE in space), contrasting them with lumped CSTR balances in time only.

Why Other Options Are Wrong:
CSTR: Uniform state by assumption → lumped parameter.On–off controller: Not a spatially distributed physical medium.None of these: Incorrect because a tubular reactor is a classic distributed example.

Common Pitfalls:
Assuming any large vessel is distributed; the key is the mixing model, not size.

Final Answer:
Tubular reactor