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?
An ideal monoatomic gas is contained inside a cylinder-piston assembly connected to a Hookean spring as shown in the figure. The piston is frictionless and massless. The spring constant is 10 kN/m. At the initial equilibrium state (shown in the figure), the spring is unstretched. The gas is expanded reversibly by adding 362.5 J of heat. At the final equilibrium state, the piston presses against the stoppers. Neglecting the heat loss to the surroundings, the final equilibrium temperature of the gas is __________ K (rounded off to the nearest integer).
The residence-time distribution (RTD) function of a reactor (in min$^{-1}$) is 
The mean residence time of the reactor is __________ min (rounded off to 2 decimal places).}
Ideal nonreacting gases A and B are contained inside a perfectly insulated chamber, separated by a thin partition, as shown in the figure. The partition is removed, and the two gases mix till final equilibrium is reached. The change in total entropy for the process is _________J/K (rounded off to 1 decimal place).
Given: Universal gas constant \( R = 8.314 \) J/(mol K), \( T_A = T_B = 273 \) K, \( P_A = P_B = 1 \) atm, \( V_B = 22.4 \) L, \( V_A = 3V_B \).
The following data is given for a ternary \(ABC\) gas mixture at 12 MPa and 308 K:
\(y_i\): mole fraction of component \(i\) in the gas mixture
\(\hat{\phi}_i\): fugacity coefficient of component \(i\) in the gas mixture at 12 MPa and 308 K
The fugacity of the gas mixture is __________ MPa (rounded off to 3 decimal places).