Question

Which one of the following fixes the volume of batch reactor for a particular conversion and production rate?

• A. Operating conditions like pressure and temperature
• B. Rate constant
• C. Density of mixture
• D. Catalyst

Hint:

Sizing relationship: - Kinetics determine the required mass or moles of reactant that must be processed per batch cycle. - Physical sizing requires converting mass to volume: \(V = \frac{\text{Mass}}{\text{Density}}\). - Therefore, the physical density of the mixture is the critical property that fixes the reactor volume.

Answer 1

The Correct Option is C

Concept: The performance equation for an ideal batch reactor expressing the reaction time ($t$) required to achieve a target fractional conversion ($X_A$) is written as: \[ t = C_{A0} \cdot \int_{0}^{X_A} \frac{dX_A}{-r_A} \] The total processing cycle time includes this reaction time ($t$) plus additional turnaround times ($t_{\text{down}}$) for discharging, cleaning, and recharging the reactor: \[ t_{\text{cycle}} = t + t_{\text{down}} \] The required mass or molar production rate determines the total amount of reactant that must be processed per batch.

Step 1: Relating batch reactor volume to production constraints.
The total volume of the batch reactor ($V$) must accommodate the mass volume of the reaction mixture handled per batch. Let us express the volume using the total mass of the batch charge ($M_{\text{total}}$) and the physical density of the reaction mixture ($\rho$): \[ V = \frac{M_{\text{total}}}{\rho} \] The required mass charge per batch ($M_{\text{total}}$) is determined by multiplying the target hourly production rate by the total cycle time ($t_{\text{cycle}}$).

Step 2: Evaluating the critical role of fluid density.
For a fixed chemical conversion ($X_A$) and a specified production rate, the total mass of reactants required per batch cycle ($M_{\text{total}}$) is uniquely fixed by the reaction kinetics. To convert this required mass into the actual physical volume ($V$) needed to size the reactor vessel, we must know the fluid density of the mixture ($\rho$). If the mixture density changes significantly during the reaction (e.g., due to temperature shifts or composition changes in gas-phase systems), the volume of the reactor must be adjusted to account for these density variations to hold the required mass charge. Therefore, the density of the mixture is the parameter that determines the physical volume of a batch reactor for a specified conversion and production rate.