Question

Non-isothermal plug flow reactor design requires solving simultaneously:

• A. Only the mole balance equation
• B. Mole balance and energy balance simultaneously
• C. Only the energy balance
• D. Mole balance and momentum balance

Hint:

For adiabatic reactors, the energy balance can sometimes be simplified to relate temperature directly to conversion:
\[ T = T_0 + \frac{(-\Delta H_{\text{rxn}}) \cdot X_A}{\sum C_{p,i}} \]
This allows the temperature term to be substituted back into the mole balance, reducing the system to a single differential equation.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks about the required set of equations that must be solved to design a non-isothermal Plug Flow Reactor (PFR).
This relates to non-isothermal reactor design in chemical reaction engineering.

Step 2: Key Formula or Approach:
For an isothermal reactor, the temperature remains constant, so the reaction rate constant (\( k \)) is constant.
This allows the mole balance equation to be solved independently.
However, in a non-isothermal reactor, the temperature varies along the length of the reactor due to the heat of reaction and heat exchange with the surroundings.

Step 3: Detailed Explanation:

• Coupling of Variables: The rate constant is highly temperature-dependent, as described by the Arrhenius equation:
\[ k(T) = k_0 \cdot e^{-E/RT} \]
This means the rate of reaction depends on both the conversion (from the mole balance) and the temperature (from the energy balance).

• The Governing Equations:
1. Mole Balance for PFR:
\[ \frac{dX_A}{dV} = \frac{-r_A(C_A, T)}{F_{A0}} \]
2. Energy Balance for PFR:
\[ \frac{dT}{dV} = \frac{U \cdot a \cdot (T_a - T) + (-r_A) \cdot (-\Delta H_{\text{rxn}})}{\sum F_i \cdot C_{p,i}} \]

• Simultaneous Solution: Because \( T \) appears in the mole balance equation (via \( k \)), and conversion \( X_A \) appears in the energy balance equation (via the flow rates \( F_i \) and the reaction rate \( -r_A \)), these two ordinary differential equations are coupled.
They cannot be solved independently and must be integrated simultaneously.


Step 4: Final Answer:
Designing a non-isothermal PFR requires solving the mole balance and energy balance equations simultaneously.