Solution: Residence Time Calculation for Lactose Hydrolysis in PFR

Given Parameters

• Reactor Volume, $\text{V_{reactor}}$ = 100 L
• Voidage, $\epsilon$ = 0.55
• Immobilized enzyme parameters:
  Michaelis constant, $\text{K'_{m}}$ = 0.72 $gl^{-1}$
  Maximum velocity, $\text{V'_{max}}$ = 18 $gl^{-1}h^{-1}$
• Inlet lactose concentration, $\text{C_{A0}}$ = 20 $gl^{-1}$
• Desired fractional conversion, $\text{X}$ = 0.90

Residence Time Calculation

For a plug flow reactor (PFR) with Michaelis-Menten kinetics, the residence time ($\tau$) is determined using the integrated design equation:

$ \tau = \frac{1}{\text{V'_{max}}} \left[ \text{K'_{m}} \ln\left(\frac{\text{C_{A0}}}{\text{C_{A}}}\right) + (\text{C_{A0}} - \text{C_{A}}) \right] $

First, calculate the outlet concentration ($\text{C_{A}}$) of lactose:

$ \text{C_{A}} = \text{C_{A0}} \times (1 - \text{X}) = 20 \, gl^{-1} \times (1 - 0.90) = 20 \, gl^{-1} \times 0.10 = 2 \, gl^{-1} $

Substitute the given and calculated values into the residence time equation:

$ \tau = \frac{1}{18 \, gl^{-1}h^{-1}} \left[ 0.72 \, gl^{-1} \times \ln\left(\frac{20 \, gl^{-1}}{2 \, gl^{-1}}\right) + (20 \, gl^{-1} - 2 \, gl^{-1}) \right] $

Simplify the expression:

$ \tau = \frac{1}{18} \left[ 0.72 \times \ln(10) + 18 \right] \, h $

Using the approximation $\ln(10) \approx 2.3026$:

$ \tau = \frac{1}{18} \left[ 0.72 \times 2.3026 + 18 \right] \, h $

$ \tau = \frac{1}{18} \left[ 1.6579 + 18 \right] \, h $

$ \tau = \frac{19.6579}{18} \, h \approx 1.092 \, h $

The calculated residence time is approximately 1.092 hours. This value is closest to the option 1.1 h.