Concept:
In process dynamics, a first-order system is structurally characterized by a single characteristic time constant (\(\tau\)). The time constant dictates the speed of the system's dynamic response to a change in input. It is derived by setting up a conservation balance equation (mass, energy, or charge) and rewriting the linear differential equation in standard time-constant form:
\[
\tau \frac{dy}{dt} + y = K x(t)
\]
Step 1: Analyzing system P (Thermometer).
A thermometer undergoes transient heat transfer. Let \(m\) be the mass of the thermometric fluid, \(C_p\) be its specific heat capacity, \(h\) be the convective heat transfer coefficient, and \(A\) be the surface area available for heat transfer. The unsteady-state energy balance equation matches heat accumulation with convective heat transfer input:
\[
m C_p \frac{dT}{dt} = h A (T_\infty - T)
\]
Rearranging into standard form:
\[
\left(\frac{m C_p}{h A}\right) \frac{dT}{dt} + T = T_\infty
\]
Comparing this to the standard first-order form, the characteristic thermal time constant is:
\[
\tau_P = \frac{m C_p}{h A} \quad \Rightarrow \quad P \rightarrow 1
\]
Step 2: Analyzing system Q (Mixing process).
Consider a continuous-flow perfectly stirred tank configuration of constant volume \(V\) with volumetric flow rate \(q\). The dynamic component balance for a tracer species leads to a total transient hold-up equation. The average residence time or mixing process time constant is determined by the total physical hold-up volume divided by the throughput volumetric flow rate:
\[
\tau_Q = \frac{V}{q} \quad \Rightarrow \quad Q \rightarrow 3
\]
Step 3: Analyzing system R (Liquid level system).
For a liquid-filled storage vessel with cross-sectional tank area \(A\) and an outlet flow resistance line \(R\), the mass balance states that the change in liquid volume over time equals the inlet flow minus the outlet flow (\(q_{out} = \frac{h}{R}\)):
\[
A \frac{dh}{dt} = q_{in} - \frac{h}{R} \quad \Rightarrow \quad (AR) \frac{dh}{dt} + h = R q_{in}
\]
Thus, the characteristic time constant of a liquid level system is the product of the tank's cross-sectional area and the outlet flow line resistance:
\[
\tau_R = AR \quad \Rightarrow \quad R \rightarrow 4
\]
Step 4: Analyzing system S (RC circuit).
For a standard electrical circuit featuring a resistor \(R\) connected in series with a capacitor \(C\), applying Kirchhoff's voltage law yields a first-order differential charging equation. The time constant is defined as the product of electrical resistance and capacitance:
\[
\tau_S = RC \quad \Rightarrow \quad S \rightarrow 2
\]
Combining these matching pairs, the correct option sequence is \(P-1, Q-3, R-4, S-2\), which corresponds exactly to Option (4).
