Dynamics of two identical tanks in series: for two first-order tanks of the same size and the same flow resistance connected in series (non-interacting), state the qualitative nature of the overall step response of the combined system.

Introduction / Context:
Two perfectly mixed process tanks in series, each behaving as a first-order lag, together form a second-order dynamic system. When both tanks have the same capacity and the same hydraulic resistance (i.e., equal time constants), the closed-loop free response to a step input shows a characteristic shape that control engineers classify as over damped, critically damped, or under damped. This question tests recognition of that qualitative behavior.

Given Data / Assumptions:

• Two non-interacting, identical first-order tanks in series.
• Each tank time constant: τ (same for both).
• Linear, constant-density mixing; no recirculation.

Concept / Approach:
The transfer function of one tank is 1/(τs + 1). Two in series give G(s) = 1/(τs + 1)^2, which is a standard second-order system with a double real pole at s = −1/τ. A double pole on the negative real axis has no oscillatory component and corresponds to an over damped response (slower, with an inflection point), not to critical damping (distinct condition ζ = 1 with complex-to-real transition) and not to under damping (which would require complex conjugate poles).

Step-by-Step Solution:

Verification / Alternative check:
The step response of 1/(τs + 1)^2 is y(t) = 1 − (1 + t/τ) * e^(−t/τ), a monotonic, sigmoid-like rise without overshoot, confirming over damping.

Why Other Options Are Wrong:

Common Pitfalls:
Assuming “second order” means oscillatory by default. Real repeated poles still yield monotone, over damped responses.

Final Answer:
Over damped