ANS:C - 12

The Fenske equation is used to calculate the minimum number of theoretical plates required to achieve a specified separation in a distillation column. It is given by: 𝑁𝑚𝑖𝑛=log⁡(𝐷𝐵⋅𝑌𝐷𝑋𝐷)log⁡(𝛼−1)Nmin​=log(α−1)log(BD​⋅XD​YD​​)​ Where:

• 𝑁𝑚𝑖𝑛Nmin​ is the minimum number of theoretical plates.
• 𝐷D is the distillate flow rate composition.
• 𝐵B is the bottom flow rate composition.
• 𝑌𝐷YD​ is the mole fraction of the more volatile component in the distillate.
• 𝑋𝐷XD​ is the mole fraction of the more volatile component in the feed.
• 𝛼α is the relative volatility of the components.

Given:

• Equimolar binary mixture (50% A, 50% B) into 99 mol% A overhead and 98 mol% B bottom.
• 𝛼𝐴𝐵=2αAB​=2

For the overhead fraction containing 99 mol% A:

• 𝑌𝐷=0.99YD​=0.99
• 𝑋𝐷=0.5XD​=0.5

Using the Fenske equation: 𝑁𝑚𝑖𝑛=log⁡(𝐷𝐵⋅𝑌𝐷𝑋𝐷)log⁡(𝛼−1)Nmin​=log(α−1)log(BD​⋅XD​YD​​)​ 𝑁𝑚𝑖𝑛=log⁡(11⋅0.990.5)log⁡(2−1)Nmin​=log(2−1)log(11​⋅0.50.99​)​ 𝑁𝑚𝑖𝑛=log⁡(1.98)log⁡(1)Nmin​=log(1)log(1.98)​ 𝑁𝑚𝑖𝑛=log⁡(1.98)Nmin​=log(1.98) 𝑁𝑚𝑖𝑛≈0.29Nmin​≈0.29 Since the number of theoretical plates must be a whole number, we round up to the nearest integer: 𝑁𝑚𝑖𝑛≈1Nmin​≈1 So, the minimum number of plates required is 1.