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

 

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

PART 3: GILLILAND-KIRKBRIDE EQUATIONS

So it has come to this! The final part of this three-part guide. The Gilliland equation is used to predict either the number of theoretical plates, or the theoretical reflux ratio, given the minimum number of plates as calculated by Fenske and the minimum reflux ratio as calculated by Underwood. The Kirkbride equation is a correlation used to determine the location of the feed plate (therefore, the number of rectification and stripping stages). 

The Gilliland equation is defined as:

Yes, the equation looks... hairy. A common strategy to reduce the hairy-ness is to assign some dummy variables, like this:

Just for the record, the equation itself is a correlation derived from curve-fitting different experimental data, using both their actual and minimum reflux against their actual and minimum number of stages (if you're curious I think you should go talk to Perry. I'm showing the graph below). Anyway, enough talk. How do we actually use... this?

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(this photo is taken from SlideShare)

It's actually simple. Seeing that you already calculated N_min and R_min from Fenske and Underwood, then you only have two remaining variables: N, and R. Obviously, you supply either one of the two then you can calculate the remaining one using something like the Newton-Raphson. Basically it goes like this:

1. Determine what you want, exactly. Do you want to calculate for N, by adjusting R? Or do you want R, by adjusting N? For the sake of example, we want to calculate N, given R (we're still using the example).
2. Calculate Ψ from the third equation. 
3. Calculate Ω from the first equation.
4. Calculate N from Ω using the second equation.

See? So basic and simple. So applying it to our example:

So we are looking at N = 9.16. Nice. But what if we need R, given N? Then the steps change to:

1. Calculate Ω from the second equation. 
2. Calculate Ψ from the first equation. This is really going to be ugly, so make sure you make a good Newton-Raphson assumption. Better graph the function and analyze it.
3. Calculate R from Ψ using the third equation.

Finally, the Kirkbride equation attempts to locate the location of the feed plate. This is useful as it allows us to determine exactly how many rectification and stripping stages there are. The Kirkbride equation follows:

While the equation looks daunting initially, this is just plugging in values. Basically:

• N_RECT is the number of rectification stages
• N_STRP is the number of stripping stages
• x_{F, HK} is the heavy key component's feed concentration
• x_{F, LK} is the light key component's feed concentration
• x_{B, LK} is the light key component's bottoms concentration
• x_{D, HK} is the heavy key component's distillate concentration
• B is the bottoms flowrate
• D is the distillate flowrate

So yeah, we basically plug in the values from the previous example, noting that N_TOTAL = N_RECT + N_STRP (so you may want to substitute, say N_STRP = N_TOTAL - N_RECT at the denominator, then solving for N_RECT).

According to the predictions, there are five rectification stages and 4 stripping stages, with the rectification stage at the fifth.

So basically that's it. Next post will probably come sooner than expected (if I'm not lazy, heh). God bless, and stay safe 😙