Question

A typical example of a physical system with second order under-damped characteristics is a:

• A. U-tube manometer
• B. Spring – loaded diaphragm valve
• C. CSTR with first order reaction
• D. Thermocouple kept immersed in a liquid filled thermal well

Hint:

The U-tube manometer is the classic textbook example of a physical second-order system capable of under-damped oscillations. Its damping coefficient \(\zeta\) is inversely proportional to the column radius and directly proportional to liquid viscosity: \(\zeta = \frac{4\mu}{R^2\rho} \sqrt{\frac{3L}{2g}}\).

Answer 1

The Correct Option is A

Concept: A second-order system can exhibit an under-damped response (\(\zeta \lt 1\)) if it contains mechanisms to store energy in two different physical forms (such as kinetic energy and potential energy) with minimal internal dissipative friction. Under-damped systems oscillate following step changes.

Step 1: Dynamic analysis of the listed devices.

• CSTR and Thermocouple in a well: These are thermal/concentration processes containing series resistances and capacities. They are modeled as multicapacity systems, resulting in an overdamped second-order response (\(\zeta \gt 1\)). They cannot oscillate naturally.

• Spring-loaded diaphragm valve: While it contains mass and a spring, it is highly dampened by process fluids and structural seals to remain critically damped or overdamped for stable throttling.

• U-tube manometer: When a pressure difference is applied across a U-tube column containing low-viscosity mercury, the liquid accelerates. The system continuously interchanges energy between the fluid kinetic energy (mass motion) and gravity potential energy (liquid head height). Because internal viscous friction is small, the mercury column bounces up and down, exhibiting clear under-damped oscillatory behavior before settling.