[CHEMICAL ENGINEERING] Shortcut distillation - FENSKE-UNDERWOOD-GILLILAND-KIRKBRIDE! (how?) PART 3: Gilliland and Kirkbride Equation

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[CHEMICAL ENGINEERING] Shortcut distillation - FENSKE-UNDERWOOD-GILLILAND-KIRKBRIDE! (how?) PART 2: Underwood Equation

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In this post we shall continue the discussion of the Fenske-Underwood-Gilliland-Kirkbride method, in particular, the Underwood equations.

PART 2: UNDERWOOD EQUATION

Anyway, an important assumption in the following calculations is that the nonkey components do not distribute (ie. nonkey components are intermediate in volatility between the keys; as after all, this is a shortcut method); as it will make computations much simpler. Also, as with the Fenske equation, I will not be discussing how the equations have been derived.

So just a little summary of results from the previous Fenske discussion here:


We use the Underwood equations to get Rmin, or the minimum reflux ratio. There are actually "two" forms of this single equation; the first form is used to calculate ∅, the the absorption factor for a reference component in the rectifying-section pinch-point zone. The second one is used to compute for the value of Rmin itself. 

The first form of the Underwood equation is defined as:


where αi with the overbar is the mean relative volatility, xFi is the feed concentration, and q is the quality of the feed (where 0 is saturated vapor and 1 is saturated liquid). We need to determine the value of ∅ in order to calculate the value of the minimum reflux ratio using the second Underwood equation, defined as:


where xDi is the distillate composition.

The remaining question now, is how to determine ∅. The most important thing to note is that it lies between the relative volatilities of the light key and the heavy key; that is:

αLK > ∅ > αHK

This is the reason why we used the heavy key's relative volatility as reference: this makes finding ∅ easier as it will be greater than 1, but less than the relative volatility of the lighter key. 

A basic algorithm for computing ∅ can take the form of:

  1. First, calculate 1 - q; then
  2. Guess some random value between αLK and 1
  3. Calculate for each component 
  4. Take the sum, this sum must be equal to 1 - q.
This is easily done in MS Excel Solver. The final converged value is ∅ = 1.43;


Next, we use this value of ∅ to calculate the value of Rmin:

Therefore, the minimum reflux ratio for the separation Rmin = 1.17.

Next we will discuss the use of the Gilliland correlation, to determine the number of actual plates given a desired reflux ratio, or vice-versa.

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