Question

For the constant-density systems, the performance equations are identical for______

• A. Batch reactor and plug flow reactor
• B. Batch reactor and back mix reactor
• C. Plug flow reactor and back mix reactor
• D. Batch reactor, plug flow reactor and back mix reactor

Hint:

The equivalence \( t = \tau \) only holds for constant-density systems.
If the fluid density changes during the reaction (variable-volume systems), the PFR space-time equation must include the expansion factor (\( \epsilon_A \)), making it mathematically different from the batch reactor equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to identify which two reactor types share mathematically identical performance equations when operating under constant-density (liquid phase or constant volume gas phase) conditions.
This is a standard comparison in chemical reaction engineering.

Step 2: Key Formula or Approach:
We compare the performance equations for a batch reactor and a plug flow reactor (PFR) for a reactant A:
1. Batch Reactor:
\[ t = C_{A0} \cdot \int_0^{X_A} \frac{dX_A}{-r_A} \]
2. Plug Flow Reactor (PFR):
\[ \tau = \frac{V}{v_0} = C_{A0} \cdot \int_0^{X_A} \frac{dX_A}{-r_A} \]

Step 3: Detailed Explanation:

• Equivalence of Equations: By comparing the two performance equations, we can see that:
\[ t = \tau \]
This means the real time \( t \) required for a reaction to reach a certain conversion in a batch reactor is identical to the space-time \( \tau \) required to reach the same conversion in a plug flow reactor, provided the feed concentration and temperature are the same.

• Physical Interpretation: In a batch reactor, reactants are charged at \( t=0 \) and react uniformly over time.
In a plug flow reactor, fluid elements behave as small, independent batch reactors sliding down the length of the tube.
Since there is no back-mixing in a PFR, the residence time of a fluid element at a distance \( z \) corresponds directly to the reaction time \( t \) in a batch reactor.

• CSTR comparison: A back-mix reactor (CSTR) has a completely different performance equation because it operates at a uniform, fully-mixed state:
\[ \tau_{\text{CSTR}} = \frac{C_{A0} \cdot X_A}{-r_A} \]
This equation is algebraic rather than integral.


Step 4: Final Answer:
For constant-density systems, the performance equations are identical for the Batch reactor and the Plug flow reactor.