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GATE 2023 CH – Question 25

Instrumentation and Process Control · Process modelling, transfer functions and dynamic response · 1 mark · Multiple choice

A liquid surge tank has $F_{in}$ and $F_{out}$ as the inlet and outlet flow rates respectively, as shown in the figure below. $F_{out}$ is proportional to the square root of the liquid level $h$. The cross-sectional area of the tank is 20 cm$^2$. Density of the liquid is constant everywhere in the system. At steady state, $F_{in} = F_{out} = 10$ cm$^3$s$^{-1}$ and $h = 16$ cm. The variation of $h$ with $F_{in}$ is approximated as a first order transfer function. Which one of the following is the CORRECT value of the time constant (in seconds) of this system?

an open tank with the level $h$, the inlet $F_{in}$ at the top and the outlet $F_{out}$ at the bottom.
  1. 20
  2. 32
  3. 64
  4. 128

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Correct answer: (C) 64

Explanation

Write $F_{out} = k\sqrt{h}$, with $k = \frac{10}{\sqrt{16}} = 2.5$. The balance is $A\frac{dh}{dt} = F_{in} - k\sqrt{h}$. Linearising about the steady state, $A\frac{dh'}{dt} = F_{in}' - \frac{k}{2\sqrt{\bar{h}}}h'$, and $\frac{k}{2\sqrt{\bar{h}}} = \frac{2.5}{8} = 0.3125$ cm²/s. The time constant is $\tau = \frac{A}{0.3125} = \frac{20}{0.3125} = 64$ s.