Liquid flowing through a heat exchanger (HX) is heated. A bypass stream is provided to control the temperature of the heated exit stream. From the given plumbing options, the one that provides the most effective temperature control for large disturbances while avoiding vaporization in the heat exchanger is

Step 1: Understanding vaporization risk.
If bypassed cold liquid mixes before entering the heat exchanger, the inlet temperature fluctuates, leading to sudden drops in heat-transfer driving force. During disturbances, this may cause overheating in parts of the HX, increasing the chance of vaporization.
Step 2: Identifying the proper configuration.
The safest and most effective arrangement is where the heat exchanger always receives the full stream and the bypass flow mixes only at the outlet. This ensures:
1. Stable HX inlet temperature,
2. Reduced risk of vaporization,
3. Better control authority because the bypassed stream directly manipulates the exit temperature.
Step 3: Selecting the correct option.
Among the options, only Diagram (C) shows the bypass line mixing with the heated stream after the HX. This provides the best temperature control with minimal risk of overheating or vapor formation inside the exchanger.
Final Answer: (C)
In the block diagram shown in the figure, the transfer function $G=\dfrac{K}{(\tau s+1)}$ with $K>0$ and $\tau>0$. The maximum value of $K$ below which the system remains stable is \(\underline{\hspace{2cm}}\) (rounded off to two decimal places).

It is required to control the volume of the contents in the jacketed reactor shown in the figure.
Which one of the following schemes can be used for feedback control?
Consider a process with transfer function: \[ G_p = \frac{2e^{-s}}{(5s + 1)^2} \] A first-order plus dead time (FOPDT) model is to be fitted to the unit step process reaction curve (PRC) by applying the maximum slope method. Let \( \tau_m \) and \( \theta_m \) denote the time constant and dead time, respectively, of the fitted FOPDT model. The value of \( \frac{\tau_m}{\theta_m} \) is __________ (rounded off to 2 decimal places).
Given: For \( G = \frac{1}{(\tau s + 1)^2} \), the unit step output response is: \[ y(t) = 1 - \left(1 + \frac{t}{\tau}\right)e^{-t/\tau} \] The first and second derivatives of \( y(t) \) are: \[ \frac{dy(t)}{dt} = \frac{t}{\tau^2} e^{-t/\tau} \] \[ \frac{d^2y(t)}{dt^2} = \frac{1}{\tau^2} \left(1 - \frac{t}{\tau}\right) e^{-t/\tau} \]
Choose the transfer function that best fits the output response to a unit step input change shown in the figure:

An electrical wire of 2 mm diameter and 5 m length is insulated with a plastic layer of thickness 2 mm and thermal conductivity \( k = 0.1 \) W/(m·K). It is exposed to ambient air at 30°C. For a current of 5 A, the potential drop across the wire is 2 V. The air-side heat transfer coefficient is 20 W/(m²·K). Neglecting the thermal resistance of the wire, the steady-state temperature at the wire-insulation interface __________°C (rounded off to 1 decimal place).

GIVEN:
Kinematic viscosity: \( \nu = 1.0 \times 10^{-6} \, {m}^2/{s} \)
Prandtl number: \( {Pr} = 7.01 \)
Velocity boundary layer thickness: \[ \delta_H = \frac{4.91 x}{\sqrt{x \nu}} \]
The first-order irreversible liquid phase reaction \(A \to B\) occurs inside a constant volume \(V\) isothermal CSTR with the initial steady-state conditions shown in the figure. The gain, in kmol/m³·h, of the transfer function relating the reactor effluent \(A\) concentration \(c_A\) to the inlet flow rate \(F\) is:
